#7658778

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

4474 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)(11,20)(12,18)(13,16)(14,19)(15,17)(21,24)(22,23)(25,26)(27,30)(28,37)(29,34)(31,35)(32,36)(33,39)(38,40);(1,2,4,5)(3,6,7,8)(9,11,16,15)(10,17,13,20)(12,22,14,21)(18,24,19,23)(25,27,31,36)(26,32,35,30)(28,34,39,40)(29,37,38,33);(9,12,16,14)(10,18,13,19)(11,21,15,22)(17,23,20,24)(25,28,31,39)(26,33,35,37)(27,34,36,40)(29,30,38,32);(1,3,4,7)(2,6,5,8)(9,13,16,10)(11,17,15,20)(12,19,14,18)(21,23,22,24)(25,29,31,38)(26,34,35,40)(27,33,36,37)(28,30,39,32);(9,14,16,12)(10,19,13,18)(11,21,15,22)(17,23,20,24)(25,30)(26,27)(28,38)(29,39)(31,32)(33,34)(35,36)(37,40);(9,15,16,11)(10,17,13,20)(12,22,14,21)(18,23,19,24)(25,26)(27,32)(28,33)(29,34)(30,36)(31,35)(37,39)(38,40)

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7565974);