Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #374018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14134 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2471 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)(3,8,15,24)(4,16)(5,7)(6,11)(9,17)(10,20)(12,23,14,21)(13,28,29,31)(18,25)(19,26)(22,30,32,27);(2,7)(3,12)(5,20)(9,16)(10,11)(13,27)(14,15)(17,18)(21,23)(25,26)(28,31)(29,30);(1,3)(2,8,11,24)(4,13)(5,14)(6,15)(7,21,10,23)(9,22,25,32)(12,20)(16,28,26,31)(17,27)(18,30)(19,29);(1,4)(2,9)(3,13)(5,18)(6,19)(7,16)(8,22)(10,26)(11,25)(12,27)(14,30)(15,29)(17,20)(21,28)(23,31)(24,32)Polycyclic presentation:
2^9|a^2=h,c^2=fh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72216029, #72216112, #72216142, … |
| maximal normal subgroup of | #72216029, #72216112, #72216142, … |
| maximal quotient of | #72237515, C2×#374018, #72338202 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281214);