#9007719

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

4464 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,e^2=hi,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, 8914915);