#474789

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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

836 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=egh,c^2=g,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, 381985);