#9835862

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

4342 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9743058);