#405137

order	512
nilpotency class	3
isoclinism	isoclinic class of #393963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2297 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)(7,9)(8,10)(11,12)(13,14)(15,20)(16,27)(17,22)(18,31)(19,23)(21,29)(24,40)(25,34)(26,36)(28,38)(30,39)(32,35)(33,41)(37,43)(42,44);(2,4)(5,7,8,11)(6,9,10,12)(13,15)(14,20)(16,24)(17,25)(18,32)(19,26)(21,37)(22,34)(23,36)(27,40)(28,41)(29,43)(30,42)(31,35)(33,38)(39,44);(13,16,19,30)(14,21,23,38)(15,24,26,42)(17,29,18,28)(20,33,36,37)(22,39,31,27)(25,43,32,41)(34,40,35,44);(6,10)(9,12)(13,17,19,18)(14,22,23,31)(15,25,26,32)(16,28,30,29)(20,34,36,35)(21,27,38,39)(24,41,42,43)(33,44,37,40)

Polycyclic presentation:

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

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, 312333);