#451094

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

1194 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)(6,9)(10,11)(13,14,16,18)(15,17,19,20)(21,22,27,30,26,32,25,31)(23,28,35,36,24,34,33,29);(5,6,8,9)(7,11,12,10)(13,15)(14,17)(16,19)(18,20)(21,23,26,24)(22,28,32,34)(25,33,27,35)(29,30,36,31);(2,4)(15,19)(17,20)(21,24,26,23)(22,29,32,36)(25,33,27,35)(28,30,34,31);(5,7,8,12)(6,10,9,11)(22,30)(25,27)(28,36)(29,34)(31,32)(33,35)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ei,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi

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, 358290);