#9923889

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

4533 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)(3,4)(5,6)(7,8)(9,10)(11,18)(12,17)(13,15)(14,16)(19,24)(20,23)(21,22)(25,26)(27,30)(28,31)(29,37)(32,34)(33,40)(35,36)(38,39);(2,6)(3,7)(9,11,14,13)(10,15,16,18)(12,19,20,21)(17,22,23,24)(25,27,32,31)(26,30,34,28)(29,35,40,39)(33,38,37,36);(10,16)(15,18)(17,23)(22,24)(25,28,32,30)(26,27,34,31)(29,38,40,36)(33,39,37,35);(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,23)(21,24)(25,29,32,40)(26,33,34,37)(27,35,31,39)(28,36,30,38);(1,3,5,7)(2,4,6,8)(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,30,32,28)(26,27,34,31)(29,38,40,36)(33,35,37,39);(1,4)(2,3)(5,8)(6,7)(9,13,14,11)(10,18,16,15)(12,19,20,21)(17,22,23,24)(25,31,32,27)(26,30,34,28)(29,39,40,35)(33,38,37,36)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9831085);