#476622

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	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1334 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,d^c=de,h^d=hi

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, 383818);