Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34681 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2029 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18,21,22)(19,23,20,24);(2,5)(6,8)(18,22)(19,23)(20,24);(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,21)(22,24);(9,12,13,11)(10,14,15,16)(17,20,21,19)(18,24,22,23);(1,3,4,7)(2,6,5,8)(19,20)(23,24)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^c=hi,e^d=ei,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72308682, #72308834 |
| maximal normal subgroup of | #72308682, #72308834 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359936);