Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5009 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,7)(9,10)(11,15)(12,14)(13,16)(18,22)(19,24);(9,11,13,12)(10,14,16,15)(17,18)(19,20)(21,22)(23,24);(5,6,8,7)(9,11,13,12)(10,15,16,14)(17,19)(18,23)(20,22)(21,24);(1,2,3,4)(5,7,8,6)(9,12,13,11)(10,15,16,14)(17,19)(18,20)(21,24)(22,23);(9,10)(11,15)(12,14)(13,16)(17,20)(18,24)(19,22)(21,23)Polycyclic presentation:
2^9|b^2=fh,c^2=fg,d^2=hi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
| maximal subgroup of | #72232214, #72232243, #72232262, … |
| maximal normal subgroup of | #72232214, #72232243, #72232262, … |
| maximal quotient of | #72330020, #72331289 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 730654);