#8924326

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

4512 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)(9,10,14,17)(11,18,13,15)(12,16,21,24)(19,23,20,22)(25,26,31,35)(27,30,37,32)(28,38,36,33)(29,39,40,34);(3,7)(6,8)(11,13)(15,18)(19,20)(22,23)(25,27,31,37)(26,32,35,30)(28,29,36,40)(33,39,38,34);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,17,18)(12,20,21,19)(16,23,24,22)(25,28,31,36)(26,33,35,38)(27,29,37,40)(30,34,32,39);(9,12,14,21)(10,16,17,24)(11,19,13,20)(15,22,18,23)(25,29,31,40)(26,34,35,39)(27,36,37,28)(30,38,32,33);(1,3)(2,6)(4,7)(5,8)(9,13,14,11)(10,15,17,18)(12,19,21,20)(16,23,24,22)(25,27)(26,30)(28,29)(31,37)(32,35)(33,34)(36,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,16)(13,15)(14,17)(19,23)(20,22)(21,24)(25,30)(26,27)(28,39)(29,38)(31,32)(33,40)(34,36)(35,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8831522);