#8600934

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

4432 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:

(9,10)(11,17)(12,14)(13,16)(15,18)(19,24)(20,22)(21,23)(25,26,31,35)(27,32,38,37)(28,39,29,33)(30,34,40,36);(2,5)(6,8)(10,12)(11,18)(16,20)(19,21)(25,27,31,38)(26,32,35,37)(28,36,29,34)(30,33,40,39);(1,2,4,5)(3,6,7,8)(10,12)(15,17)(16,20)(23,24)(25,28,31,29)(26,33,35,39)(27,34,38,36)(30,37,40,32);(3,7)(6,8)(9,11,14,18)(10,15,12,17)(13,21,22,19)(16,24,20,23)(25,29,31,28)(26,33,35,39)(27,36,38,34)(30,37,40,32);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,17,18,15)(13,20,22,16)(19,24,21,23)(25,26)(27,37)(28,39)(29,33)(30,34)(31,35)(32,38)(36,40);(1,3)(2,6)(4,7)(5,8)(9,13,14,22)(10,16,12,20)(11,19,18,21)(15,23,17,24)(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,e^2=g,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, 8508130);