#8393713

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

4473 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,19,22,16)(12,13,18,17)(20,21,23,24)(25,26,30,34)(27,28,32,31)(29,33,37,40)(35,36,39,38);(1,2)(3,7)(4,5)(6,8)(10,15)(12,18)(16,19)(20,23)(25,27,30,32)(26,31,34,28)(29,35,37,39)(33,38,40,36);(2,5)(6,7)(10,15)(13,17)(16,19)(21,24)(29,37)(33,40)(35,39)(36,38);(3,8)(6,7)(9,11,14,22)(10,16,15,19)(12,23,18,20)(13,21,17,24)(25,26)(27,28)(29,33)(30,34)(31,32)(35,36)(37,40)(38,39);(1,3,4,8)(2,6,5,7)(9,12,14,18)(10,17,15,13)(11,20,22,23)(16,21,19,24)(25,28)(26,32)(27,34)(29,36)(30,31)(33,39)(35,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,13,14,17)(10,18,15,12)(11,21,22,24)(16,20,19,23)(25,29,30,37)(26,33,34,40)(27,35,32,39)(28,36,31,38)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300909);