#8406952

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

4889 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8314148);