#9203854

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

4510 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)(3,7)(4,5)(6,8)(9,10,14,12)(11,15,18,16)(13,17,21,20)(19,22,24,23)(25,26)(27,35)(28,30)(29,32)(31,34)(33,40)(36,37)(38,39);(2,5)(6,7)(9,11)(10,15)(12,16)(13,19)(14,18)(17,22)(20,23)(21,24)(25,26,31,34)(27,32,29,35)(28,30,33,40)(36,39,38,37);(1,2)(3,6)(4,5)(7,8)(11,18)(15,16)(19,24)(22,23)(25,27,31,29)(26,32,34,35)(28,38,33,36)(30,37,40,39);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,15)(13,19)(14,18)(17,23)(20,22)(21,24)(25,28,31,33)(26,30,34,40)(27,36,29,38)(32,39,35,37);(9,12,14,10)(11,16,18,15)(13,17,21,20)(19,22,24,23)(25,29,31,27)(26,32,34,35)(28,36,33,38)(30,37,40,39);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30)(26,33)(27,37)(28,34)(29,39)(31,40)(32,36)(35,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9111050);