Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2537 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)(6,9)(7,11)(13,14,18,19)(15,20,25,28)(16,26,24,21)(17,22,23,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,27)(17,26)(18,25)(19,28)(21,23)(22,24);(1,2,3,4)(13,16)(14,21)(15,23)(17,25)(18,24)(19,26)(20,22)(27,28);(5,7,8,11)(6,10,9,12)(13,14,18,19)(15,20,25,28)(16,21,24,26)(17,27,23,22);(6,9)(10,12)(13,17)(14,22)(15,24)(16,25)(18,23)(19,27)(20,21)(26,28)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,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.
| maximal subgroup of | #72309061 |
| maximal normal subgroup of | #72309061 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590988);