Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,18,23,27)(16,19,24,28)(21,25,22,26);(6,9)(10,12)(13,15)(14,18)(16,21)(17,23)(19,25)(20,27)(22,24)(26,28);(1,2,3,4)(15,21)(16,24)(18,25)(19,28)(22,23)(26,27);(15,22)(16,24)(18,26)(19,28)(21,23)(25,27);(5,7,8,11)(6,10,9,12)(15,23)(18,27)(21,22)(25,26)Polycyclic presentation:
2^9|a^2=i,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72257277, #72257411, #72257498, … |
| maximal normal subgroup of | #72257277, #72257411, #72257498, … |
| maximal quotient of | #72264383, #72264411, #72308429 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352060);