#232224

order	512
nilpotency class	3
isoclinism	isoclinic class of #232215
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×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1501 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)(7,10)(8,11)(9,12)(13,16)(14,17)(15,19)(18,20)(21,22)(23,31)(24,27)(25,29)(26,30)(28,34)(32,36)(33,35);(5,7,9,14)(6,10,12,17)(8,13,15,20)(11,18,19,16)(22,27)(23,32)(24,26)(29,30)(33,34)(35,36);(2,4)(5,8)(6,11)(7,13)(9,15)(10,16)(12,19)(14,20)(17,18)(21,23,25,34)(22,28,29,31)(24,32,26,33)(27,35,30,36);(22,29)(27,30)(28,31)(35,36)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=gi,c^b=cg,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, 139420);