Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #180873 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×#97 |
| prime radical | C2×#41 | M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(18,22,23,28)(19,25)(24,31,30,29)(26,32);(17,18,19,24,20,22,26,29,21,23,25,30,27,28,32,31);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16);(3,9)(6,12)(10,15)(13,16)Polycyclic presentation:
2^9|a^2=e,b^2=f,e^2=g,f^2=h,h^2=i,b^a=bh,f^a=fi,e^b=ei,d^c=dg110 conjugacy classes — view the full character table.
| maximal subgroup of | #72216668, #72216670, #72216771, … |
| maximal normal subgroup of | #72216668, #72216670, #72216771, … |
| maximal quotient of | #72217246, D4×#1313, C2×#479977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387173);