#6557068

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fi,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,e^d=ehi

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, 6464264);