#7709438

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

4453 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=g,e^2=h,f^2=g,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, 7616634);