#8534634

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

4776 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)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,26,31,33)(27,32,29,28)(30,34,39,40)(35,38,37,36);(2,3)(6,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,27,31,29)(26,32,33,28)(30,35,39,37)(34,38,40,36);(1,3,5,2)(4,7,8,6)(11,17)(14,21)(18,23)(22,24)(26,33)(27,29)(34,40)(35,37);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,28,31,32)(26,27,33,29)(30,38,39,36)(34,37,40,35);(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,29,31,27)(26,32,33,28)(30,37,39,35)(34,38,40,36);(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,39)(32,38)(33,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=hi,d^2=h,e^2=gi,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, 8441830);