#8600918

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

4343 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8508114);