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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1951 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)(14,19)(15,23)(16,17)(21,22)(25,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,14,16,19,18,21,17,22)(15,24,26,28,25,20,23,27);(2,4)(6,9)(10,12)(13,15,16,26,18,25,17,23)(14,20,19,27,21,24,22,28);(5,7)(6,10)(8,11)(9,12)(15,25)(20,24)(23,26)(27,28);(7,11)(10,12)(14,21)(19,22)(20,24)(27,28)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=f,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, 6459981);