Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418470 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2676 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15376 |
| prime radical | #23406 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1288 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)(4,15,17,8)(5,19)(6,10)(9,21)(11,14)(12,22,16,25)(13,23)(18,26,32,29)(20,24)(27,31,30,28);(1,3)(2,7)(4,16)(5,20)(6,13)(8,25)(9,11)(10,23)(12,17)(14,21)(15,22)(18,28)(19,24)(26,30)(27,29)(31,32);(1,2)(3,11)(4,8)(5,19)(6,10)(7,20)(9,21)(12,27)(13,24)(14,23)(15,17)(16,30)(18,29)(22,31)(25,28)(26,32);(1,4)(2,8)(3,12,13,16)(5,18)(6,17)(7,22,23,25)(9,26)(10,15)(11,27,24,30)(14,28,20,31)(19,29)(21,32)Polycyclic presentation:
2^9|a^2=e,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72338482, #72338485, #72338495 |
| maximal normal subgroup of | #72338482, #72338485, #72338495 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335134);