#441603

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
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

1116 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)(7,10)(8,12)(9,11)(13,14,16,18)(15,17,19,20)(22,27)(25,34)(28,36)(29,31)(30,35)(32,33);(5,7,9,8)(6,10,11,12)(13,15,16,19)(14,17,18,20)(21,22,26,30)(23,31,24,28)(25,35,34,27)(29,32,36,33);(5,8,9,7)(6,10,11,12)(15,19)(17,20)(21,23,26,24)(22,28,30,31)(25,32,34,33)(27,36,35,29);(2,4)(21,24,26,23)(22,29,30,36)(25,32,34,33)(27,31,35,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,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=de,e^b=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348799);