#9783295

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

4317 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=gh,e^2=g,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, 9690491);