#6432614

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2788 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi

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, 6339810);