Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,11)(12,15,16,14)(17,18)(19,21)(20,22)(23,24);(2,5)(6,8)(9,11,13,10)(12,15,16,14)(17,19)(18,21)(20,24)(22,23);(9,12)(10,14)(11,15)(13,16)(17,18)(19,23)(20,22)(21,24);(12,16)(14,15)(19,24)(21,23);(18,22)(21,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72279424, #72279724, #72280721, … |
| maximal normal subgroup of | #72279424, #72279724, #72280721, … |
| maximal quotient of | #72280110, #72280199, #72281347, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7574439);