#433314

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

990 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=ei,d^2=e,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, 340510);