#406420

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

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

1380 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=f,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,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, 313616);