#8821859

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

4429 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8729055);