#8228661

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4688 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)(3,4)(5,6)(7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26)(27,32)(28,34)(29,37)(30,36)(31,33)(35,40)(38,39);(2,6)(3,8)(10,14)(15,17)(16,21)(22,23)(25,27,31,30)(26,32,33,36)(28,35,39,37)(29,34,40,38);(2,6)(4,7)(12,20)(16,21)(18,24)(22,23)(26,33)(27,30)(34,38)(35,37);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,31,39)(26,34,33,38)(27,37,30,35)(29,36,40,32);(1,3)(2,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29,31,40)(26,35,33,37)(27,38,30,34)(28,36,39,32);(1,4,5,7)(2,3,6,8)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,27)(26,36,33,32)(28,35,39,37)(29,34,40,38)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8135857);