Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | #30468 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1572 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,16)(13,14)(19,25)(24,27)(26,30)(31,32);(1,3,4,8)(2,5,6,7)(18,22)(19,26)(20,28)(23,29)(25,30)(31,32);(10,14)(15,16)(17,18)(19,27)(20,29)(21,23)(22,28)(24,25)(26,32)(30,31);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(18,23)(22,29)(24,27)(31,32);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(17,19)(18,24)(20,30)(21,25)(22,31)(23,27)(26,28)(29,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=g,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308624, #72308967 |
| maximal normal subgroup of | #72308624, #72308967 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6638112);