Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518936 |
| 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 | #25364 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,9,8)(7,10,12,11)(15,22)(16,25)(17,24)(19,26)(20,27)(23,28);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,14)(15,23)(16,26)(17,27)(18,21)(19,25)(20,24)(22,28);(5,8,9,6)(7,11,12,10)(13,15,24,25,18,16,17,22)(14,19,27,23,21,28,20,26);(7,12)(10,11)(13,16,24,22,18,15,17,25)(14,19,27,23,21,28,20,26);(1,3,4,2)Polycyclic presentation:
2^9|a^2=ghi,b^2=f,c^2=hi,d^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72308472, #72308768 |
| maximal normal subgroup of | #72308472, #72308768 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511378);