Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,8)(9,12)(10,11)(13,14,18,20)(15,19,17,16)(22,24)(26,27);(5,6,9,12)(7,8,11,10)(13,15,18,17)(14,16,20,19)(21,22,25,24)(23,27,28,26);(5,7,9,11)(6,10,12,8)(13,16,18,19)(14,15,20,17)(21,23,25,28)(22,26,24,27);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,16)(17,19)(18,20);(5,8,9,10)(6,11,12,7)(13,17,18,15)(14,19,20,16)(21,24,25,22)(23,27,28,26)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72326975, #72326993, #72327043, … |
| maximal normal subgroup of | #72326975, #72326993, #72327043, … |
| maximal quotient of | #72330795, #72335941 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752884);