#7004692

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1539 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6911888);