Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #403107 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2547 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14868 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1213 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,11,6,10)(3,13,21,24,15,8,16,25)(4,17,19,28,20,9,22,27)(14,29,31,32,30,23,18,26);(1,3)(2,8,11,13)(4,18)(5,21)(6,16)(7,15)(9,26,17,32)(10,24,12,25)(14,19)(20,31)(22,30)(23,27,29,28);(1,4)(2,9)(3,14)(5,22)(6,19)(7,20)(8,23)(10,27)(11,17)(12,28)(13,29)(15,30)(16,31)(18,21)(24,26)(25,32);(2,10,11,12)(8,24,13,25)(9,27,17,28)(23,26,29,32)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72338514, #72338519, #72338527 |
| maximal normal subgroup of | #72338514, #72338519, #72338527 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310303);