#416636

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1780 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323832);