#232733

order	512
nilpotency class	3
isoclinism	isoclinic class of #232724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1179 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,6,10)(3,11,14,15)(4,9,5,8)(7,13,16,12)(17,18,23,28)(19,29,33,39)(20,25,36,44)(21,26,22,27)(24,32,40,31)(30,45,46,41)(34,38,47,48)(35,42,37,43);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,15,9,11)(17,19,23,33)(18,24,28,40)(20,34,36,47)(21,32,22,31)(25,41,44,45)(26,29,27,39)(30,37,46,35)(38,43,48,42);(2,8)(3,12)(4,5)(9,10)(11,15)(13,14)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,42)(27,43)(28,44)(29,45)(31,34)(32,47)(33,46)(39,41)(40,48);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,39,40,29)(25,42,44,43)(30,34,46,47)(38,41,48,45)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dhi,e^b=eh,g^b=gh,e^c=ei,f^c=fi,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, 139929);