Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | |
2343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(15,22)(17,20)(24,26)(25,27);(5,7)(6,9)(8,12)(10,11)(13,14)(15,24)(16,21)(17,27)(18,23)(19,28)(20,25)(22,26);(1,2,3,4)(13,15)(14,20)(16,25)(17,23)(18,22)(19,27)(21,26)(24,28);(6,10)(9,11)(13,16,18,19)(14,21,23,28)(15,25,22,27)(17,24,20,26);(7,12)(9,11)(13,17)(14,22)(15,23)(16,26)(18,20)(19,24)(21,25)(27,28)Polycyclic presentation:
2^9|c^2=gi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589484);