Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499953 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3205 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #24208 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1189 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)(3,13)(4,17)(5,8)(6,23)(7,12)(9,18)(10,16)(11,15)(14,28)(19,24)(20,26)(21,27)(22,25)(29,31)(30,32);(1,3,6,5,7,16,15,22)(2,8,11,13,12,25,23,10)(4,14,19,18,21,32,20,31)(9,26,28,27,29,24,30,17);(1,4)(2,9)(3,14)(5,18)(6,19)(7,21)(8,24)(10,27)(11,28)(12,29)(13,17)(15,20)(16,32)(22,31)(23,30)(25,26);(1,5,6,16,7,22,15,3)(2,10,11,8,12,13,23,25)(4,18,19,32,21,31,20,14)(9,27,28,24,29,17,30,26)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fh,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219591, #72219593, #72219594, … |
| maximal normal subgroup of | #72219591, #72219593, #72219594, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407880);