#9719191

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

4587 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)(5,8)(6,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,33)(28,32)(29,31)(30,35)(34,40)(36,38)(37,39);(1,2)(3,6)(4,7)(5,8)(11,17)(15,21)(18,23)(22,24)(25,27)(26,30)(28,36)(29,35)(31,33)(32,40)(34,37)(38,39);(1,2,5,8)(3,4,7,6)(10,14)(15,21)(16,19)(22,24)(26,31)(30,33)(32,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,26)(27,30)(28,32)(29,31)(33,35)(34,38)(36,40)(37,39);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28,29,37)(26,32,31,39)(27,34,35,36)(30,38,33,40);(1,4)(2,7)(3,8)(5,6)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,27)(26,30)(28,36)(29,35)(31,33)(32,40)(34,37)(38,39)

Polycyclic presentation:

2^9|c^2=i,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9626387);