#7643905

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

4552 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7551101);