#7649949

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

4588 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)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24)(25,26,31,33)(27,34,37,30)(28,38,29,32)(35,40,36,39);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,27)(26,30)(28,35)(29,36)(31,37)(32,40)(33,34)(38,39);(11,18)(15,22)(19,23)(20,24)(25,28,31,29)(26,32,33,38)(27,35,37,36)(30,40,34,39);(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24)(25,26)(27,30)(28,32)(29,38)(31,33)(34,37)(35,40)(36,39);(1,3,5,4)(2,7,6,8)(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)(25,29,31,28)(26,32,33,38)(27,36,37,35)(30,40,34,39);(1,4,5,3)(2,8,6,7)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,30)(26,27)(28,39)(29,40)(31,34)(32,36)(33,37)(35,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7557145);