#477933

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
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

970 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=f,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, 385129);