Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#449 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,25)(21,31)(22,32)(23,27)(24,29)(26,30);(1,2,3,4)(5,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19,20,30,23,21,22,29)(18,24,32,31,27,26,25,28);(6,8)(10,14)(13,16)(17,20,23,22)(18,25,27,32)(19,29,21,30)(24,31,26,28);(2,4)(17,21,23,19)(18,26,27,24)(20,30,22,29)(25,28,32,31)Polycyclic presentation:
2^9|b^2=efi,c^2=i,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^d=ei,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72264490, #72264515, #72264552, … |
| maximal normal subgroup of | #72264490, #72264515, #72264552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328213);