#8635456

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

4434 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=g,e^2=gh,f^2=hi,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, 8542652);