#232803

order	512
nilpotency class	3
isoclinism	isoclinic class of #232771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1651 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|d^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=di,e^b=eh,g^b=gh,d^c=dh,e^c=ei,f^c=fhi,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139999);