#416168

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1376 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eh,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, 323364);