#8103934

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4591 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8011130);