#8088192

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

4306 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)(11,17,13,16)(12,14,18,24)(20,21,22,23)(25,26,31,35)(27,32,37,39)(28,38,29,33)(30,34,36,40);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,19,17)(12,22,18,20)(14,21,24,23)(25,27,31,37)(26,32,35,39)(28,30,29,36)(33,40,38,34);(11,13)(16,17)(20,22)(21,23)(25,28,31,29)(26,33,35,38)(27,36,37,30)(32,34,39,40);(1,2,5,4)(3,6,8,7)(9,12)(10,14)(11,20)(13,22)(15,18)(16,23)(17,21)(19,24)(25,29,31,28)(26,33,35,38)(27,30,37,36)(32,34,39,40);(1,4,5,2)(3,7,8,6)(9,13,15,11)(10,17,19,16)(12,20,18,22)(14,23,24,21)(25,26)(27,32)(28,38)(29,33)(30,34)(31,35)(36,40)(37,39);(9,14)(10,18)(11,21)(12,19)(13,23)(15,24)(16,20)(17,22)(25,30)(26,34)(27,29)(28,37)(31,36)(32,33)(35,40)(38,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=gi,e^2=g,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, 7995388);