#234806

order	512
nilpotency class	3
isoclinism	isoclinic class of #232373
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1281 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)(5,6,9,13)(7,11,14,20)(8,16,18,12)(10,17,19,15)(21,22,26,30)(23,28,32,35)(24,29,25,33)(27,34,36,31);(5,7)(6,10)(8,17)(9,14)(11,16)(12,20)(13,19)(15,18)(21,23,26,32)(22,27,30,36)(24,31,25,34)(28,29,35,33);(2,4)(5,7)(6,11)(8,15)(9,14)(10,16)(12,19)(13,20)(17,18)(21,23)(22,28)(24,31)(25,34)(26,32)(27,29)(30,35)(33,36);(7,14)(10,19)(11,20)(15,17)(21,24,26,25)(22,29,30,33)(23,31,32,34)(27,35,36,28)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,d^b=dhi,e^b=eh,d^c=dhi,e^c=ei

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, 142002);