#472131

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

1056 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)(7,9)(8,10)(11,12)(14,16)(18,20)(21,22,26,28,25,29,24,30)(23,27,31,36,33,34,32,35);(5,7,8,11)(6,9,10,12)(13,14,17,16)(15,18,19,20)(21,23,26,31,25,33,24,32)(22,27,28,36,29,34,30,35);(2,4)(13,15,17,19)(14,18,16,20)(22,28)(23,31)(24,26)(27,34)(29,30)(32,33);(6,10)(9,12)(13,16,17,14)(15,18,19,20)(22,29)(27,34)(28,30)(35,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=gh,c^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,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, 379327);