#72285992

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#10788
commutator	C₂²×C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₄×C₈	C₂³
prime radical	C₂×#3169	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1377 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,3,8,5,11,13,7,6,9,4,10,14,16,12,15)(17,18)(19,27,30,23)(20,24)(21,25)(22,26)(28,39,42,34)(29,40,43,35)(31,36)(32,37)(33,38)(41,47,48,45)(44,46);(2,7)(3,12)(4,13)(5,14)(8,11)(9,15)(10,16)(17,19,22,30)(18,23,26,27)(20,28,32,42)(21,29,33,43)(24,34,37,39)(25,35,38,40)(31,41,44,48)(36,45,46,47);(2,8,9,10)(3,13)(4,12)(5,14)(7,11,15,16)(17,20,21,31,22,32,33,44)(18,24,25,36,26,37,38,46)(19,28,29,41,30,42,43,48)(23,34,35,45,27,39,40,47)

Polycyclic presentation:

2^10|a^2=e,b^2=h,c^2=d,d^2=h,e^2=i,f^2=i,g^2=j,i^2=j,b^a=bf,c^a=cegj,d^a=dj,e^b=ehij,e^c=ei,f^a=fh,f^b=fij,f^c=fi,g^b=ghj,g^c=gj,i^b=ij,i^c=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286019, #72343274, #72343326
maximal normal subgroup of	#72286019, #72343274, #72343326
maximal quotient of	#72286349, #72286437

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);