#147121

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₈×D₄
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,4)(5,6,8,7)(9,10)(11,16)(12,26)(13,20)(14,19)(15,21)(17,27)(18,29)(22,36)(23,33)(24,32)(25,34)(28,35)(30,37)(31,39)(38,40);(2,4)(9,11,14,24,13,23,17,30,15,25,26,36,29,39,35,40)(10,16,19,32,20,33,27,37,21,34,12,22,18,31,28,38);(5,7,8,6)(9,12)(10,17)(11,22)(13,28)(14,18)(15,19)(16,30)(20,26)(21,35)(23,38)(24,31)(25,32)(27,29)(33,36)(34,40)(37,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,e^c=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54317);