#435575

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

876 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)(3,4)(5,6,9,11)(7,10,8,12)(13,14,16,18)(15,17,19,20)(21,22)(23,30)(24,26)(25,29)(27,31)(28,33)(32,34)(35,36);(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,19)(18,20)(22,26)(23,31)(24,33)(28,29)(30,34)(32,36);(5,8,9,7)(6,10,11,12)(15,19)(17,20)(23,32)(27,35)(30,34)(31,36);(2,4)(21,23,33,36,25,32,24,31)(22,27,26,34,29,35,28,30)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=f,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,d^c=di

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, 342771);