#9847642

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

4332 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)(3,7)(4,5)(6,8)(9,10,15,18)(11,19,21,16)(12,22,23,14)(13,17,24,20)(25,26,31,28)(27,34,29,32)(30,36,38,33)(35,39,37,40);(2,5)(6,7)(9,11,15,21)(10,16,18,19)(12,20,23,17)(13,22,24,14)(25,26,31,28)(27,32,29,34)(30,33,38,36)(35,39,37,40);(1,2)(3,6)(4,5)(7,8)(25,27,31,29)(26,32,28,34)(30,35,38,37)(33,39,36,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,14)(11,20)(13,16)(15,23)(17,21)(18,22)(19,24)(25,28,31,26)(27,32,29,34)(30,36,38,33)(35,39,37,40);(9,13,15,24)(10,17,18,20)(11,14,21,22)(12,16,23,19)(25,29,31,27)(26,32,28,34)(30,37,38,35)(33,39,36,40);(9,14)(10,12)(11,13)(15,22)(16,20)(17,19)(18,23)(21,24)(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=h,d^2=h,e^2=ghi,f^2=h,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, 9754838);