#8534762

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

4772 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=i,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8441958);