#472724

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

798 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=efgh,c^2=f,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379920);