#7709591

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

4481 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7616787);