#466000

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1060 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)(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,20)(17,19)(21,22,26,31)(23,29,24,28)(25,30,27,32)(33,36,34,35);(6,9)(10,12)(13,15,17,16)(14,18,19,20)(21,23,25,33,26,24,27,34)(22,28,30,35,31,29,32,36);(5,7)(6,10)(8,11)(9,12)(14,19)(18,20)(21,24,26,23)(22,29,31,28)(25,33,27,34)(30,35,32,36);(2,4)(7,11)(10,12)(13,16,17,15)(14,20,19,18)(21,25,26,27)(22,30,31,32)(23,33,24,34)(28,35,29,36)

Polycyclic presentation:

2^9|a^2=e,b^2=fg,c^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,g^c=gi

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, 373196);