#8993594

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

4267 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)(9,10,14,13)(11,15,20,18)(12,16,21,22)(17,23,19,24)(26,30)(27,33)(28,36)(37,40);(3,8)(6,7)(9,11)(10,15)(12,19)(13,18)(14,20)(16,24)(17,21)(22,23)(25,26,31,30)(27,34,33,29)(28,32,36,38)(35,40,39,37);(1,3,5,8)(2,6,4,7)(9,12,14,21)(10,16,13,22)(11,17,20,19)(15,23,18,24)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,39)(38,40);(3,8)(6,7)(9,13,14,10)(11,18,20,15)(12,16,21,22)(17,23,19,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,36)(33,39)(34,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,19)(13,22)(14,21)(15,24)(17,20)(18,23)(25,29,31,34)(26,33,30,27)(28,37,36,40)(32,39,38,35);(1,4,5,2)(3,6,8,7)(9,11)(10,15)(12,19)(13,18)(14,20)(16,24)(17,21)(22,23)(25,30,31,26)(27,34,33,29)(28,38,36,32)(35,40,39,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8900790);