#232781

order	512
nilpotency class	3
isoclinism	isoclinic class of #232755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1309 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:

(1,2)(3,6)(4,7)(5,8)(9,10)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,29)(22,38)(23,33)(25,32)(27,37)(28,36)(30,35)(31,40)(34,39);(2,6)(3,4)(7,8)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,34)(18,21)(19,32)(20,33)(22,28)(26,31)(29,40)(30,39)(35,38)(36,37);(1,2)(3,6)(4,7)(5,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,26)(19,35)(20,36)(21,37)(23,39)(24,27)(25,40)(32,34)(33,38);(1,3,5,4)(2,6,8,7)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,30,29)(16,21,33,32)(17,26,36,35)(22,39,40,27)(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,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,d^b=di,e^b=eh,g^b=gh,e^c=ei,f^c=fhi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139977);