#72287689

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#106
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#3444	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #60204 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2185 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=h,c^2=f,e^2=gj,f^2=h,g^2=i,i^2=j,b^a=bh,c^a=chj,d^a=dj,d^b=dh,d^c=dj,e^a=ej,e^b=ehj,e^c=eg,f^e=fj,g^c=gi,i^c=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287792, #72319058, #72319144
maximal normal subgroup of	#72287792, #72319058, #72319144

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.8);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.7*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);