#232388

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	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1239 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)(5,6,10,13)(7,15,17,20)(8,18,19,12)(9,16,11,14)(21,22,27,31)(23,32,33,36)(24,34,35,28)(25,30,26,29);(5,7)(6,8)(9,18)(10,17)(11,12)(13,19)(14,20)(15,16)(21,23,27,33)(22,24,31,35)(25,28,26,34)(29,32,30,36);(2,4)(5,8)(6,12)(7,16)(9,15)(10,19)(11,20)(13,18)(14,17)(21,24)(22,28)(23,30)(25,32)(26,36)(27,35)(29,33)(31,34);(6,13)(7,17)(12,18)(14,16)(21,25,27,26)(22,29,31,30)(23,28,33,34)(24,32,35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139584);