#9851043

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

4516 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)(11,16)(12,17)(13,21)(14,15)(18,23)(19,20)(22,24)(25,26,29,32)(27,33,35,30)(28,31,37,40)(34,39,36,38);(2,5)(6,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,22)(20,24)(21,23)(25,27)(26,30)(28,36)(29,35)(31,39)(32,33)(34,37)(38,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(18,21)(19,24)(25,27,29,35)(26,30,32,33)(28,34,37,36)(31,38,40,39);(1,3,4,8)(2,6,5,7)(9,11,14,17)(10,16,15,12)(13,19,23,24)(18,22,21,20)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,37)(39,40);(9,12,14,16)(10,17,15,11)(13,22,23,20)(18,19,21,24)(25,28)(26,31)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36);(9,13)(10,18)(11,19)(12,20)(14,23)(15,21)(16,22)(17,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=gh,e^2=gi,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, 9758239);