#7740540

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

4307 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7647736);