#232329

order	512
nilpotency class	3
isoclinism	isoclinic class of #232313
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1328 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=hi,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,d^b=di,e^b=eh,g^b=gh,e^c=ei,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, 139525);