#476582

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, 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

1158 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=ei,c^2=e,d^2=i,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, 383778);