#7717283

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

4404 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=ghi,f^2=g,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, 7624479);