Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,19)(17,26)(18,22)(21,24)(23,27)(25,28);(6,9)(10,12);(14,19)(15,17)(16,24)(20,21)(22,28)(23,25)(26,27);(5,7,8,11)(6,10,9,12)(13,15,18,17)(14,20,22,26)(16,23,25,24)(19,27,28,21)Polycyclic presentation:
2^9|a^2=g,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^c=di,f^c=fh,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72213483, #72225017, #72225106, … |
| maximal normal subgroup of | #72213483, #72225017, #72225106, … |
| maximal quotient of | #72224991, #72286524, C2×#494030 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401226);