#246170

order	512
nilpotency class	3
isoclinism	isoclinic class of #246159
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	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1382 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=eh,f^b=fi,e^c=ei,e^d=eh,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153366);