#232414

order	512
nilpotency class	3
isoclinism	isoclinic class of #232393
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	#2656	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1308 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)(7,16)(8,13)(9,20)(10,14)(11,15)(12,17)(18,19)(21,22)(23,31)(24,30)(25,29)(26,28)(27,33)(32,35)(34,36);(5,7)(6,12)(8,15)(9,13)(10,17)(11,18)(14,19)(16,20)(21,23,26,32)(22,27,28,36)(24,34,25,33)(29,35,30,31);(2,4)(5,8)(6,13)(7,14)(9,19)(10,16)(11,12)(15,17)(18,20)(21,22)(23,31)(24,30)(25,29)(26,28)(27,33)(32,35)(34,36);(5,9,11,10)(6,14,15,20)(7,13,18,17)(8,16,12,19)(22,28)(23,32)(29,30)(33,34)

Polycyclic presentation:

2^9|b^2=i,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=di,e^b=eh,g^b=gh,d^c=dh,e^c=ei,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139610);