#9720498

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

4514 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)(3,7)(4,5)(6,8)(9,10,13,14)(11,15,19,17)(12,20,21,16)(18,23,24,22)(25,26,30,31)(27,32,35,28)(29,37,39,33)(34,36,40,38);(2,5)(6,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(26,31)(28,32)(33,37)(36,38);(1,2)(3,6)(4,5)(7,8)(10,14)(15,17)(16,20)(22,23)(25,27,30,35)(26,28,31,32)(29,34,39,40)(33,36,37,38);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,38)(36,37)(39,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,28,30,32)(26,27,31,35)(29,38,39,36)(33,40,37,34);(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,29,30,39)(26,33,31,37)(27,34,35,40)(28,36,32,38)

Polycyclic presentation:

2^9|a^2=h,c^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9627694);