Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471449 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16797 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1105 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(4,17)(5,6)(9,22)(11,21)(12,25)(13,18)(14,15)(19,20)(23,27)(26,29)(30,31);(1,2,6,21,7,11,5,8)(3,12,14,25,16,22,15,9)(4,10,19,27,20,28,17,23)(13,29,31,32,30,26,18,24);(1,3)(2,9)(4,18)(5,15)(6,14)(7,16)(8,22)(10,26)(11,25)(12,21)(13,19)(17,30)(20,31)(23,32)(24,27)(28,29);(1,4)(2,10)(3,13)(5,17)(6,19)(7,20)(8,23)(9,24)(11,28)(12,29)(14,31)(15,18)(16,30)(21,27)(22,26)(25,32)Polycyclic presentation:
2^9|b^2=egh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72339678, #72339679, #72339684 |
| maximal normal subgroup of | #72339678, #72339679, #72339684 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378645);