#7782719

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

4524 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)(11,16)(12,18)(13,22)(14,19)(15,17)(20,23)(21,24)(25,26)(27,35)(28,33)(29,34)(30,36)(31,37)(32,39)(38,40);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,19)(13,21)(14,18)(15,20)(17,23)(22,24)(25,27)(26,32)(28,36)(29,38)(30,34)(31,39)(33,40)(35,37);(10,17)(11,20)(12,22)(14,21)(25,28,31,29)(26,33,37,34)(27,38,39,36)(30,35,40,32);(1,3,5,4)(2,7,6,8)(9,12,15,22)(10,18,17,13)(11,19,20,24)(14,23,21,16)(25,29,31,28)(26,34,37,33)(27,38,39,36)(30,35,40,32);(9,13,15,18)(10,12,17,22)(11,21,20,14)(16,19,23,24)(25,27)(26,35)(28,38)(29,36)(30,34)(31,39)(32,37)(33,40);(1,4,5,3)(2,8,6,7)(9,14,15,21)(10,19,17,24)(11,18,20,13)(12,23,22,16)(25,30)(26,36)(27,34)(28,32)(29,35)(31,40)(33,39)(37,38)

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=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, 7689915);