#473960

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=ei,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,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, 381156);