#8077315

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

4587 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7984511);