#8750340

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

4581 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8657536);