#7645484

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

4561 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)(11,18)(12,21)(13,16)(14,22)(15,17)(19,20)(23,24)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,36)(39,40);(1,2)(3,7)(4,8)(5,6)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(1,3,5,4)(2,7,6,8)(9,11,15,13)(10,16,17,18)(12,19,22,23)(14,20,21,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(1,3,5,4)(2,7,6,8)(9,12)(10,14)(11,19)(13,23)(15,22)(16,20)(17,21)(18,24)(25,28,30,34)(26,31,33,27)(29,36,38,40)(32,39,37,35);(9,13,15,11)(10,16,17,18)(12,19,22,23)(14,24,21,20)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,38)(39,40);(1,4,5,3)(2,8,6,7)(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,22)(18,23)(25,29,30,38)(26,32,33,37)(27,35,31,39)(28,36,34,40)

Polycyclic presentation:

2^9|c^2=ghi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,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, 7552680);