#8731002

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

4437 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8638198);