Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1845 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(18,22)(19,25)(20,28)(23,27)(26,31)(30,32);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,18,21,22)(19,23,27,25)(20,29,28,24)(26,32,30,31);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,25,21,27,18,23)(20,30,29,31,28,26,24,32);(2,6)(3,8)(11,16)(13,15)(17,20,21,28)(18,24,22,29)(19,26,27,30)(23,31,25,32);(1,4,5,7)(2,3,6,8)(10,14)(13,15)(19,27)(20,28)(23,25)(24,29)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=g,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72308682, #72308971 |
| maximal normal subgroup of | #72308682, #72308971 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6820000);