#8088006

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

4249 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7995202);