Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394043 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2613 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14918 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9)(3,11)(4,16)(5,19)(7,22,21,10)(8,15)(12,24,30,27)(13,17,31,28)(14,25)(18,20)(23,26)(29,32);(1,3)(2,8,9,11)(4,17,19,28)(5,13)(6,15)(7,14)(10,25,22,29)(12,18,30,23)(16,31)(20,27)(21,32)(24,26);(1,4)(2,5)(3,12)(6,19)(7,23)(8,24)(9,16)(10,26)(11,27)(13,29)(14,28)(15,30)(17,32)(18,21)(20,22)(25,31);(1,5)(2,4)(3,13)(6,16)(7,20)(8,17)(9,19)(10,18)(11,28)(12,29)(14,27)(15,31)(21,26)(22,23)(24,32)(25,30)Polycyclic presentation:
2^9|a^2=ef,b^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,e^c=ei,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72338511, #72338517, #72338526 |
| maximal normal subgroup of | #72338511, #72338517, #72338526 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312967);