#6854859

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1435 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)(9,10,13,12)(11,14,16,15)(17,18)(19,28)(20,25)(21,26)(22,35)(23,27)(24,31)(29,39)(30,43)(32,40)(33,42)(34,41)(36,37)(38,46)(44,48)(45,47);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,14,12,15)(19,29)(24,37)(28,39)(30,44)(31,36)(38,47)(43,48)(45,46);(9,10,13,12)(11,15,16,14)(17,19)(18,24)(20,30)(21,31)(22,36)(23,29)(25,38)(26,39)(27,37)(28,35)(32,45)(33,44)(34,46)(40,43)(41,48)(42,47);(2,6)(5,7)(17,20,23,33)(18,25,27,42)(19,30,29,44)(21,34,22,32)(24,38,37,47)(26,41,35,40)(28,43,39,48)(31,46,36,45);(3,8)(5,7)(9,12,13,10)(11,14,16,15)(17,21,23,22)(18,26,27,35)(19,31,29,36)(20,32,33,34)(24,39,37,28)(25,40,42,41)(30,45,44,46)(38,43,47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=i,e^2=h,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, 6762055);