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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1927 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)(7,11)(8,12)(9,10);(2,3)(6,10)(11,12)(13,14,18,22)(15,23,25,19)(16,21,17,27)(20,26,28,24);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,15)(14,19)(16,26)(17,24)(18,25)(20,21)(22,23)(27,28);(1,3,4,2)(13,15)(14,20)(16,24)(17,26)(18,25)(19,21)(22,28)(23,27);(5,8,9,7)(6,11,10,12)(13,16,18,17)(14,21,22,27)(15,24,25,26)(19,28,23,20)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=g,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7471303);