#437526

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1186 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)(5,6,9,8)(7,11,12,10)(13,14,16,18)(15,19,20,17)(21,22,24,26)(23,30,31,27)(25,28,33,29)(32,35,36,34);(1,3,4,2)(5,7)(6,10)(8,11)(9,12)(13,15,16,20)(14,17,18,19)(22,26)(27,30)(28,29)(34,35);(2,3)(5,6,9,8)(7,10,12,11)(14,18)(17,19)(21,23,24,31)(22,27,26,30)(25,32,33,36)(28,34,29,35);(2,3)(5,8,9,6)(7,10,12,11)(15,20)(17,19)(22,28)(25,33)(26,29)(27,34)(30,35)(32,36)

Polycyclic presentation:

2^9|a^2=eg,b^2=fi,c^2=efi,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 344722);