#6556895

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1377 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)(3,6)(4,5)(7,8)(9,10)(11,19)(12,21)(13,17)(14,16)(15,18)(20,23)(22,24)(25,26)(27,29)(28,36)(30,33)(31,38)(32,34)(35,37)(39,40);(2,5)(6,8)(9,11,15,14)(10,16,18,19)(12,22,13,20)(17,24,21,23)(25,27,34,33,30,32,29,26)(28,36,40,31,37,35,38,39);(1,3,4,7)(2,6,5,8)(9,12,15,13)(10,17,18,21)(11,20,14,22)(16,23,19,24)(25,28,29,38,30,37,34,40)(26,31,32,36,33,39,27,35);(9,13,15,12)(10,17,18,21)(11,20,14,22)(16,24,19,23)(28,37)(31,39)(35,36)(38,40);(9,14,15,11)(10,16,18,19)(12,22,13,20)(17,23,21,24)(25,29,30,34)(26,27,33,32)(28,38,37,40)(31,35,39,36)

Polycyclic presentation:

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

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, 6464091);