Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C22×#121 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
1406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,9)(7,10)(11,12)(13,14)(15,17,19,20)(16,18)(21,22,23,25,24,27,28,26);(5,6,10,12,8,11,7,9)(13,15)(14,17,18,20)(16,19)(22,25,27,26);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,d^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72279344, #72279350, #72279441 |
| maximal normal subgroup of | #72279344, #72279350, #72279441 |
| maximal quotient of | #72279581, #72279658 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6931);