#9847989

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

4524 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=g,e^2=gi,f^2=g,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, 9755185);