#8901523

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

4332 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8808719);