#403743

order	512
nilpotency class	3
isoclinism	isoclinic class of #396101
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 310939);