#8253898

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

4473 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,19)(18,23)(22,24)(25,26,31,32)(27,33,28,30)(29,37,39,34)(35,38,36,40);(2,6)(3,7)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(26,32)(27,28)(34,37)(35,36);(1,3,5,7)(2,4,6,8)(10,14)(15,17)(16,20)(22,23)(25,27,31,28)(26,30,32,33)(29,35,39,36)(34,38,37,40);(1,4)(2,7)(3,6)(5,8)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,28,31,27)(26,33,32,30)(29,36,39,35)(34,40,37,38);(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29,31,39)(26,34,32,37)(27,35,28,36)(30,38,33,40);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,30,31,33)(26,27,32,28)(29,38,39,40)(34,35,37,36)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8161094);