#499176

order	512
nilpotency class	4
isoclinism	isoclinic class of #498795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1062 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)(3,4)(9,10)(11,23)(12,30)(13,21)(14,20)(15,34)(16,32)(17,19)(18,25)(22,33)(24,26)(27,40)(28,39)(29,31)(35,37)(36,38);(5,6)(7,8)(9,11,17,29)(10,18,19,37)(12,31,21,23)(13,26,30,27)(14,25,15,35)(16,36,33,28)(20,38,34,39)(22,24,32,40);(2,4)(6,8)(9,12,13,19,33,34,15,22,17,21,30,10,16,20,14,32)(11,24,25,39,36,37,27,31,29,40,35,38,28,18,26,23);(10,19)(12,21)(18,37)(20,34)(22,32)(23,31)(24,40)(38,39)

Polycyclic presentation:

2^9|b^2=i,c^2=e,e^2=hi,f^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,e^b=eh,f^b=fhi,g^b=ghi,h^b=hi,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, 406372);