Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #32913 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(19,27)(23,31)(24,32)(28,29);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,18,21,25)(19,24,27,32)(20,26,22,30)(23,29,31,28);(9,12,13,11)(10,14,15,16)(17,19)(18,23)(20,28)(21,27)(22,29)(24,30)(25,31)(26,32);(2,6)(5,7)(17,19)(18,24)(20,29)(21,27)(22,28)(23,30)(25,32)(26,31);(3,8)(5,7)(17,20)(19,28)(21,22)(24,32)(26,30)(27,29)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285044, #72308849, #72308985, … |
| maximal normal subgroup of | #72285044, #72308849, #72308985, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6599860);