#232809

order	512
nilpotency class	3
isoclinism	isoclinic class of #232809
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

1298 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)(4,8)(5,7)(9,10)(11,20)(12,25)(13,30)(14,19)(15,18)(16,23)(17,29)(21,38)(22,32)(24,33)(26,37)(27,35)(28,36)(31,39)(34,40);(2,6)(3,4)(7,8)(9,11)(10,16)(12,26)(13,24)(14,23)(15,22)(17,34)(18,33)(19,32)(20,30)(21,28)(25,31)(27,40)(29,39)(35,38)(36,37);(1,2)(3,6)(4,8)(5,7)(9,12)(10,17)(11,21)(13,28)(14,29)(15,27)(16,31)(18,36)(19,35)(20,37)(22,39)(23,26)(24,40)(25,30)(32,34)(33,38);(2,7)(6,8)(10,18)(11,22)(12,27)(19,30)(23,24)(28,29)(31,38)(34,37)

Polycyclic presentation:

2^9|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=dhi,e^b=eh,g^b=gh,d^c=dh,e^c=ei,f^c=fhi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140005);