Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9980 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1144 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1144 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) = 128 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2)(3,4)(5,6)(7,12,10,8)(9,11)(13,14)(15,24,27,35)(16,28)(17,22)(18,33)(19,23)(20,26,36,32)(21,30)(25,37,42,34)(29,38)(31,39)(40,43,44,41);(2,4)(5,7,9,10)(6,8,11,12)(13,15)(14,20)(16,25)(17,32)(18,26)(19,27)(21,37)(22,35)(23,36)(24,33)(28,44)(29,43)(30,41)(31,42)(34,38)(39,40);(5,8)(6,10)(7,11)(9,12)(13,16,19,31)(14,21,23,38)(15,25,27,42)(17,30,18,29)(20,34,36,37)(22,39,33,28)(24,40,35,44)(26,43,32,41)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dhi,e^c=ei,g^c=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18520);