Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #390446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14572 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,14,7)(4,15,18,24)(5,9)(6,10)(8,17,25,20)(12,29)(13,26,19,23)(16,32)(21,27)(22,30)(28,31);(1,3)(2,7)(4,16)(5,19)(6,14)(8,22)(9,26)(10,11)(12,18)(13,21)(15,29)(17,30)(20,31)(23,27)(24,32)(25,28);(1,4)(2,8)(3,12,14,16)(5,20)(6,18)(7,22,11,28)(9,24)(10,25)(13,31,19,30)(15,27)(17,21)(23,29,26,32);(2,9)(5,21)(7,23)(8,24)(10,27)(11,26)(13,19)(15,25)(17,20)(22,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=h,c^2=egh,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72338616, #72338621, #72338636 |
| maximal normal subgroup of | #72338616, #72338621, #72338636 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297642);