#232368

order	512
nilpotency class	3
isoclinism	isoclinic class of #232351
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

1374 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)(5,6,10,14)(7,15,18,12)(8,13,9,11)(16,19,17,20)(21,22,26,30)(23,31,34,35)(24,29,25,28)(27,33,36,32);(6,11)(7,16)(8,9)(13,14)(17,18)(19,20)(21,23,26,34)(22,27,30,36)(24,33,25,32)(28,35,29,31);(2,4)(5,7)(6,12)(8,16)(9,17)(10,18)(11,19)(13,20)(14,15)(23,32)(24,25)(27,31)(28,29)(33,34)(35,36);(5,8,10,9)(6,13,14,11)(7,16,18,17)(12,20,15,19)

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=dhi,e^b=eh,g^b=gh,e^c=ei,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139564);