#416625

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	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1748 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)(15,24,27,36)(16,21,29,40)(17,31,18,23)(20,32,35,26)(25,37,41,44)(28,43,42,34)(30,39,33,38);(6,9)(10,11)(13,15,18,32,19,27,17,26)(14,20,31,36,22,35,23,24)(16,28,33,41,29,42,30,25)(21,37,39,34,40,44,38,43);(1,2,3,4)(13,16)(14,21)(15,25)(17,30)(18,33)(19,29)(20,34)(22,40)(23,38)(24,37)(26,42)(27,41)(28,32)(31,39)(35,43)(36,44);(5,7,8,12)(6,10,9,11)(14,22)(16,29)(20,35)(23,31)(24,36)(25,41)(28,42)(30,33)

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,c^b=cg,g^b=gi,d^c=di,g^c=gi

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, 323821);