#402419

order	512
nilpotency class	3
isoclinism	isoclinic class of #393909
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

1164 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=efi,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309615);