Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C26 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,22,18,20,16,21)(15,23,24,27,25,28,26,19);(1,2)(3,4)(6,9)(10,12);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,27)(21,23)(22,28);(14,20)(19,27)(21,22)(23,28);(13,16,18,17)(14,21,20,22)(15,24,25,26)(19,23,27,28)Polycyclic presentation:
2^9|a^2=g,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72237210, #72237244, #72237253, … |
| maximal normal subgroup of | #72237210, #72237244, #72237253, … |
| maximal quotient of | #72238961, #72264365, #72316136 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339373);