#413196

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2372 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320392);