#234703

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1372 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)(7,14,16,19)(8,12,9,17)(11,18,20,15)(21,22,26,30)(23,28,33,36)(24,29,25,27)(31,34,32,35);(5,7,10,16)(6,11,13,20)(8,15,9,18)(12,19,17,14)(22,27)(24,25)(28,35)(29,30)(31,32)(34,36);(2,4)(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)(21,23,26,33)(22,28,30,36)(24,32,25,31)(27,34,29,35);(21,24,26,25)(22,29,30,27)(23,31,33,32)(28,34,36,35)

Polycyclic presentation:

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

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, 141899);