Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | |
2251 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)(7,10)(8,9)(11,12)(13,14,18,16)(15,19,24,25)(17,20,27,26)(21,22,28,23);(6,9)(7,11)(13,15)(14,19)(16,25)(17,22)(18,24)(20,28)(21,26)(23,27);(1,2,3,4)(13,16,18,14)(15,21,24,28)(17,20,27,26)(19,22,25,23);(6,9)(10,12)(14,16)(19,25)(20,26)(21,28);(5,7,8,11)(6,10,9,12)(15,22)(17,27)(19,28)(20,26)(21,25)(23,24)Polycyclic presentation:
2^9|a^2=i,c^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281541, #72282270, #72282353, … |
| maximal normal subgroup of | #72281541, #72282270, #72282353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592812);