#8103923

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

4587 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,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, 8011119);