Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419458 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2709 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15375 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1239 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)(4,10)(5,21)(6,14)(7,12)(8,13)(9,16)(15,30)(17,28)(18,23)(19,29)(20,27)(22,26)(24,31)(25,32);(1,3)(2,9)(4,15)(5,6)(7,16)(8,17)(10,24)(11,14)(12,28)(13,21)(18,19)(20,32)(22,31)(23,27)(25,26)(29,30);(1,4)(2,10)(3,15)(5,18)(6,19)(7,22)(8,20)(9,23)(11,25)(12,26)(13,27)(14,29)(16,31)(17,32)(21,24)(28,30);(1,5,6,17,8,16,7,3)(2,11,12,9,13,28,14,21)(4,18,19,32,20,31,22,15)(10,25,26,23,27,30,29,24)Polycyclic presentation:
2^9|d^2=ef,e^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,d^b=de,e^b=ei,f^b=fi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318864, #72318890, #72318910, … |
| maximal normal subgroup of | #72318864, #72318890, #72318910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335067);