#423936

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

1281 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:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=f,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^d=hi

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, 331132);