#6466543

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1972 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=gh,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373739);