Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431707 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23391 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(15,23)(16,17)(19,26)(21,22)(24,25)(27,28);(1,2)(3,4)(6,9)(10,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,26)(24,27)(25,28);(2,4)(13,15)(14,19)(16,23)(17,25)(18,24)(20,27)(21,26)(22,28);(5,7,8,11)(6,10,9,12)(14,20)(19,27)(21,22)(26,28);(13,16,18,17)(14,21,20,22)(15,23,24,25)(19,26,27,28)Polycyclic presentation:
2^9|d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72307351, #72307439, #72315271, … |
| maximal normal subgroup of | #72307351, #72307439, #72315271, … |
| maximal quotient of | #72316259, #72316261 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339230);