#7658170

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

4557 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:

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

Polycyclic presentation:

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

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, 7565366);