#8105836

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
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

4430 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi

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, 8013032);