#8075484

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

4302 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,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, 7982680);