#357886

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2538 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)(9,10,14,19,15,18,13,29)(11,20,23,34,24,16,22,31)(12,25,28,33,26,40,27,17)(21,35,38,30,36,39,37,32);(5,6)(7,8)(9,11,15,24)(10,16,18,20)(12,21,26,36)(13,23,14,22)(17,32,33,30)(19,34,29,31)(25,39,40,35)(27,38,28,37);(2,4)(6,8)(9,12)(10,17,18,33)(11,21)(13,27)(14,28)(15,26)(16,30,20,32)(19,25,29,40)(22,37)(23,38)(24,36)(31,39,34,35);(12,26)(17,33)(21,36)(25,40)(27,28)(30,32)(35,39)(37,38)

Polycyclic presentation:

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

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, 265082);