#8105117

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

4419 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=hi,e^2=h,f^2=g,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, 8012313);