#409776

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

1018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=fh,c^2=ei,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^c=di

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, 316972);