Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #327 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C22×#97 | C22 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(14,20)(15,25)(16,27)(17,18)(21,24)(22,28)(23,26);(6,9,11,10)(13,14,16,21,18,24,27,20,19,23,25,28,17,22,15,26)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,e^2=h,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,h^a=hi,c^b=cg,g^b=gi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72211371, #72213167, #72213261 |
| maximal normal subgroup of | #72211371, #72213167, #72213261 |
| maximal quotient of | #72210220, #72210666, #72210669 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 590);