#446489

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

1108 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)(6,10)(11,12)(21,22,24,28)(23,30,31,26)(25,27,33,29)(32,35,36,34);(5,6)(7,11)(8,12)(9,10)(13,14,16,18)(15,19,20,17)(21,23,24,31)(22,26,28,30)(25,32,33,36)(27,34,29,35);(5,7,9,8)(6,11,10,12)(13,15,16,20)(14,17,18,19)(22,27)(25,33)(26,34)(28,29)(30,35)(32,36);(2,4)(5,8,9,7)(6,11,10,12)(23,31)(26,30)(32,36)(34,35)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=egi,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^c=fi

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, 353685);