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 | #38034 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(14,20)(15,25)(17,18)(22,26)(23,24)(27,28);(5,6)(7,10)(8,9)(11,12)(13,14,17,20,19,24,18,23)(15,21,22,28,26,16,25,27);(2,4)(6,9)(10,12)(13,15,17,22,19,26,18,25)(14,21,20,28,24,16,23,27);(5,7)(6,10)(8,11)(9,12)(13,16,19,21)(14,22,24,25)(15,20,26,23)(17,27,18,28);(7,11)(10,12)(15,26)(16,21)(22,25)(27,28)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,d^c=di,e^c=ei,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6447128);