Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C2×#102 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 446 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(2,4)(5,6,7,9,8,11,12,10)(13,14)(15,23)(16,28)(17,20)(18,32)(19,26)(21,31)(22,29)(24,36)(25,38)(27,44)(30,33)(34,41)(35,40)(37,43)(39,42);(6,9,11,10)(13,15,24,40,18,27,16,26,17,25,31,43,29,42,30,41)(14,19,33,44,22,35,21,23,20,34,28,39,32,37,36,38);(1,2,3,4)(14,20)(15,25)(21,36)(22,32)(26,41)(27,42)(28,33)(40,43)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=ei,d^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci104 conjugacy classes — view the full character table.
| maximal subgroup of | #72210554, #72210570 |
| maximal normal subgroup of | #72210554, #72210570 |
| maximal quotient of | #72210445, #72210451 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47720);