Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #38914 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(14,20)(15,24)(17,18)(21,25)(22,23)(27,28);(5,6)(7,10)(8,9)(11,12)(13,14,17,20,19,22,18,23)(15,16,21,27,25,26,24,28);(2,4)(6,9)(10,12)(13,15,18,24,19,25,17,21)(14,16,23,28,22,26,20,27);(5,7)(6,10)(8,11)(9,12)(13,16,19,26)(14,21,22,24)(15,20,25,23)(17,27,18,28);(7,11)(10,12)(14,22)(16,26)(20,23)(27,28)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6459987);