Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1846 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)(3,4)(10,13)(14,16)(17,18,20,26)(19,27,28,23)(21,24,22,25)(29,32,30,31);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,32)(26,27);(2,4)(6,8)(9,11,12,15)(10,14,13,16)(18,24)(21,22)(23,31)(25,26)(27,32)(29,30);(10,13)(14,16)(18,25)(19,28)(21,22)(23,31)(24,26)(27,32)Polycyclic presentation:
2^9|a^2=i,c^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72264477, #72264491, #72264505, … |
| maximal normal subgroup of | #72264477, #72264491, #72264505, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325271);