#10324532

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

4512 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10231728);