#8517361

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

4773 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:

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

Polycyclic presentation:

2^9|b^2=hi,d^2=g,e^2=g,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, 8424557);