Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| 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 | #35165 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,12)(9,11)(13,14,17,23,18,22,19,20)(15,24,26,27,16,21,25,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(14,20)(15,25)(16,26)(17,19)(21,24)(22,23);(2,4)(5,8,9,7)(6,10,11,12)(13,15,19,25,18,16,17,26)(14,21,20,27,22,24,23,28);(2,4)(13,16,17,25,18,15,19,26)(14,21,23,28,22,24,20,27);(14,22)(20,23)(21,24)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=fg,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284383 |
| maximal normal subgroup of | #72284383 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354364);