#8581528

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

4478 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=hi,e^2=hi,f^2=i,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, 8488724);