#8103927

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

4489 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)(3,5)(4,7)(6,8)(9,10,15,12)(11,19,21,16)(13,17,14,18)(20,24,22,23)(25,26,29,31)(27,33,34,30)(28,36,37,32)(35,38,40,39);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,19)(13,22)(14,20)(15,21)(17,24)(18,23)(25,26)(27,30)(28,36)(29,31)(32,37)(33,34)(35,39)(38,40);(2,3)(6,7)(9,12,15,10)(11,19,21,16)(13,18,14,17)(20,24,22,23)(25,27,29,34)(26,30,31,33)(28,35,37,40)(32,38,36,39);(2,3)(6,7)(10,12)(16,19)(17,18)(23,24)(26,31)(27,34)(32,36)(35,40);(1,4,5,8)(2,6,3,7)(9,13,15,14)(10,17,12,18)(11,20,21,22)(16,23,19,24)(25,28,29,37)(26,32,31,36)(27,35,34,40)(30,38,33,39);(9,14,15,13)(10,18,12,17)(11,20,21,22)(16,23,19,24)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,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, 8011123);