#8104425

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

4537 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=i,e^2=g,f^2=h,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, 8011621);