#72287174

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	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623099	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2406 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:

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

Polycyclic presentation:

2^10|a^2=j,c^2=f,d^2=gj,f^2=h,g^2=i,i^2=j,c^a=ch,d^a=dj,d^b=dhj,d^c=dg,e^d=egj,f^d=fj,g^c=gi,g^e=gij,i^c=ij,i^e=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287278, #72287726, #72287811
maximal normal subgroup of	#72287278, #72287726, #72287811

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