#7717665

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

4406 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7624861);