#9838180

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

4467 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=gh,e^2=gi,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, 9745376);