Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #199540 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60750 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,13,18,29,3,15)(4,20,23,31,22,28,19,26)(6,14,16,8,7,10,17,11)(9,27,25,24,12,32,30,21);(2,8,14,13)(3,16)(4,21)(5,17)(9,25,20,31)(10,15,29,11)(12,30,28,26)(19,23)(22,24)(27,32);(2,9)(3,17)(4,22)(5,16)(8,25)(10,28)(11,30)(12,29)(13,31)(14,20)(15,26)(19,27)(21,24)(23,32);(2,10)(8,15)(9,28)(11,13)(12,20)(14,29)(25,26)(30,31)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fhi,g^c=gh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72213216, #72213569, #72213659, … |
| maximal normal subgroup of | #72213216, #72213569, #72213659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106948);