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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,10)(8,12)(9,11)(13,14)(15,22)(16,18)(17,21)(19,25)(20,27)(23,26)(24,28);(1,2,4,3)(5,7,9,12)(6,10,11,8)(13,15)(14,19)(16,23)(17,25)(18,24)(20,26)(21,22)(27,28);(5,8,9,10)(6,7,11,12)(13,16)(14,20)(15,23)(17,27)(18,21)(19,26)(22,24)(25,28);(6,11)(7,12)(14,21)(15,24)(18,20)(19,28)(22,23)(25,26);(1,3,4,2)(14,21)(16,27)(19,22)(23,28)Polycyclic presentation:
2^9|b^2=fg,c^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72307556, #72307674, #72308463 |
| maximal normal subgroup of | #72307556, #72307674, #72308463 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7449324);