#478434

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

970 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)(10,13)(11,15)(17,18,20,22)(19,21,23,24)(25,26,28,30)(27,33,34,40)(29,37,36,32)(31,38,39,35);(1,3,4,7)(2,5,6,8)(9,10,12,13)(11,14,15,16)(17,19)(18,21)(20,23)(22,24)(27,34)(31,39)(33,40)(35,38);(2,6)(5,8)(9,11,12,15)(10,14,13,16)(19,23)(21,24)(26,30)(31,39)(32,37)(33,40);(3,7)(5,8)(11,15)(14,16)(25,27,28,34)(26,31,30,39)(29,38,36,35)(32,40,37,33)

Polycyclic presentation:

2^9|a^2=efg,b^2=fi,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,d^c=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, 385630);