#232392

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

1303 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)(7,14)(8,10)(9,19)(11,13)(12,15)(16,20)(17,18)(21,22)(23,31)(24,27)(25,29)(26,30)(28,34)(32,36)(33,35);(5,7,10,16)(6,12,8,18)(9,17,11,15)(13,20,19,14)(22,27)(23,32)(24,26)(29,30)(33,34)(35,36);(2,4)(5,8,10,6)(7,14,16,20)(9,19,11,13)(12,15,18,17)(21,23,25,34)(22,28,29,31)(24,32,26,33)(27,35,30,36);(5,9,10,11)(6,13,8,19)(7,15,16,17)(12,14,18,20)(22,29)(27,30)(28,31)(35,36)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=hi,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, 139588);