#7712934

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

4523 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)(12,20)(16,21)(18,24)(22,23)(25,26,31,33)(27,36,39,32)(28,34,40,30)(29,37,38,35);(1,3,5,8)(2,6,4,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27)(26,32)(28,37)(29,34)(30,38)(31,39)(33,36)(35,40);(1,2,5,4)(3,6,8,7)(10,14)(15,17)(16,21)(22,23)(26,33)(28,40)(32,36)(35,37);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28)(26,34)(27,37)(29,32)(30,33)(31,40)(35,39)(36,38);(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,29,31,38)(26,35,33,37)(27,34,39,30)(28,32,40,36);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,34)(26,28,33,40)(27,38,39,29)(32,37,36,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7620130);