#8390911

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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=gi,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, 8298107);