#8091720

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

4468 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)(11,17)(12,18)(13,19)(14,16)(15,21)(20,23)(22,24)(25,26)(27,28)(29,31)(30,33)(32,34)(35,36)(37,38)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,20)(13,18)(14,21)(16,17)(19,22)(23,24)(25,27)(26,32)(28,29)(30,35)(31,34)(33,39)(36,37)(38,40);(1,2,5,4)(3,6,8,7)(10,16)(13,23)(15,17)(18,24)(26,29)(27,34)(30,38)(36,39);(9,11)(10,17)(12,19)(13,18)(14,21)(15,16)(20,22)(23,24)(25,28)(26,27)(29,34)(30,39)(31,32)(33,40)(35,37)(36,38);(9,12)(10,18)(11,20)(13,15)(14,22)(16,24)(17,23)(19,21)(25,29,31,26)(27,28,34,32)(30,37,38,33)(35,36,40,39);(1,4,5,2)(3,7,8,6)(9,13,14,23)(10,19,16,20)(11,18,21,24)(12,15,22,17)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,e^2=i,f^2=gi,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, 7998916);