Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185503 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3513 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,26,22,27,20,25)(19,24,28,23,30,31,29,32);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,26)(25,31)(27,32);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,29)(21,28)(22,30)(23,26)(25,32)(27,31);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,22,21)(18,25,27,26)(19,28,30,29)(23,31,32,24)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,g^b=ghi,d^c=di,g^c=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72268383, #72268393, #72268404, … |
| maximal normal subgroup of | #72268383, #72268393, #72268404, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92881);