#476475

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(6,9)(10,12)(13,14,16,18)(15,17,19,20)(21,22)(23,31)(24,29)(25,28)(26,30)(27,34)(32,35)(33,36);(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,19)(18,20)(21,23,25,34)(22,27,28,31)(24,33,26,32)(29,35,30,36);(5,7,8,11)(6,10,9,12)(14,18)(17,20)(22,28)(23,32)(24,26)(27,35)(31,36)(33,34);(2,4)(6,9)(7,11)(15,19)(17,20)(21,24,25,26)(22,29,28,30)(23,33,34,32)(27,35,31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383671);