#435422

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=ef,c^2=fi,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=dei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 342618);