#395576

order	512
nilpotency class	3
isoclinism	isoclinic class of #393904
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2169 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)(5,6)(7,8)(9,10,13,19,15,20,14,18)(11,21,24,16,23,33,25,32)(12,26,28,36,30,35,29,17)(22,37,39,40,38,31,27,34);(1,2,3,4)(9,11,13,24,15,23,14,25)(10,16,18,21,20,32,19,33)(12,27,28,22,30,39,29,38)(17,34,35,31,36,40,26,37);(9,12,15,30)(10,17,20,36)(11,22,23,38)(13,29,14,28)(16,31,32,37)(18,26,19,35)(21,34,33,40)(24,27,25,39);(6,8)(11,23)(16,32)(21,33)(22,38)(24,25)(27,39)(31,37)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302772);