Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9383 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1148 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6)(7,9)(8,11)(10,12)(13,14)(15,18,22,25)(16,19)(17,20)(21,27,28,24)(23,26);(6,9,11,10)(13,15)(14,18,20,25)(16,21)(17,22)(19,24,26,27)(23,28);(1,2,3,4)(13,16,17,23)(14,19,20,26)(15,21,22,28)(18,24,25,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efi,d^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=di,e^b=ei,f^b=fi,d^c=di98 conjugacy classes — view the full character table.
| maximal subgroup of | #72227434, C2×#110561, #72239058, … |
| maximal normal subgroup of | #72227434, C2×#110561, #72239058, … |
| maximal quotient of | #72227430, C2×#110561, #72268608 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17757);