#6372118

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2080 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)(7,10)(8,9)(11,12)(13,14)(15,20)(16,27)(17,22)(18,32)(19,23)(21,29)(24,38)(25,34)(26,36)(28,37)(30,41)(31,35)(33,39)(40,43)(42,44);(1,3,4,2)(6,9)(10,12)(13,15,19,26)(14,20,23,36)(16,24,30,40)(17,31,18,25)(21,33,37,44)(22,35,32,34)(27,38,41,43)(28,42,29,39);(13,16,17,28,19,30,18,29)(14,21,32,41,23,37,22,27)(15,24,31,42,26,40,25,39)(20,33,34,43,36,44,35,38);(5,7,8,11)(6,10,9,12);(13,17,19,18)(14,22,23,32)(15,25,26,31)(16,28,30,29)(20,34,36,35)(21,27,37,41)(24,39,40,42)(33,43,44,38)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279314);