#234519

order	512
nilpotency class	3
isoclinism	isoclinic class of #232291
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	#9914	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1661 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,14)(8,12)(9,17)(10,13)(11,15)(16,19)(18,20)(21,22)(23,26)(24,27)(25,28)(29,34)(30,33)(31,35)(32,36);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17)(26,33)(34,35);(2,4)(21,23)(22,26)(24,31)(25,30)(27,35)(28,33)(29,32)(34,36);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,20,14)(21,24,25,32)(22,27,28,36)(23,29,30,31)(26,34,33,35)

Polycyclic presentation:

2^9|d^2=hi,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,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, 141715);