#6858812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1479 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)(3,7)(4,6)(5,8)(17,18,22,27)(19,23,28,36)(20,25,31,39)(21,24,33,26)(29,35,41,46)(30,37,42,34)(32,38,44,40)(43,47,48,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,29)(21,30)(22,28)(24,37)(25,35)(26,34)(27,36)(31,41)(32,43)(33,42)(38,47)(39,46)(40,45)(44,48);(10,14)(15,16)(18,24)(19,28)(21,33)(23,34)(25,38)(26,27)(29,41)(32,44)(35,45)(36,37)(39,40)(46,47);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,32)(22,31)(23,35)(24,38)(26,40)(27,39)(28,41)(30,43)(33,44)(34,45)(36,46)(37,47)(42,48);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(20,31)(25,39)(29,41)(32,44)(35,46)(38,40)(43,48)(45,47)

Polycyclic presentation:

2^9|a^2=i,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6766008);