Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1455 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)(3,7)(4,8)(5,11)(6,12)(9,14)(10,13)(15,21)(16,32)(17,25)(18,24)(19,27)(20,30)(22,31)(23,29)(26,28);(2,7)(3,13)(4,16)(8,22)(9,12)(10,11)(14,28)(15,30)(17,18)(20,24)(21,32)(23,25)(26,27)(29,31);(2,8)(3,14)(7,20)(9,21)(10,22)(11,24)(12,25)(13,28)(15,30)(23,26)(27,32)(29,31);(2,9)(3,13)(4,17)(7,12)(8,21)(10,27)(11,26)(14,28)(15,31)(16,18)(20,25)(22,32)(23,24)(29,30)Polycyclic presentation:
2^9|e^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72210564, #72338782 |
| maximal normal subgroup of | #72210564, #72338782 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 153563);