#6403361

order	512
nilpotency class	3
isoclinism	isoclinic class of #62265
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6310557);