#359066

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1936 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=h,e^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266262);