Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14,17,18)(15,19,23,22)(16,24,26,20)(21,28,25,27);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,17,23)(14,19,18,22)(16,25,26,21)(20,28,24,27);(6,9)(10,11)(13,16)(14,20)(15,21)(17,26)(18,24)(19,27)(22,28)(23,25);(5,7,8,12)(6,10,9,11)(14,18)(15,22)(19,23)(20,24)(21,28)(25,27);(1,3,4,2)(15,23)(16,26)(19,22)(20,24)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72307674, #72307846, #72308054 |
| maximal normal subgroup of | #72307674, #72307846, #72308054 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7458633);