Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432833 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#525 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,22)(15,23,25,28)(16,21,17,20)(19,26,27,24);(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,23)(21,28)(22,27);(2,4)(7,11)(9,12);(6,10)(9,12);(13,16,18,17)(14,20,22,21)(15,24,25,26)(19,23,27,28)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,c^b=ch,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72282916, #72282979, #72282996, … |
| maximal normal subgroup of | #72282916, #72282979, #72282996, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340033);