Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #20338 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C2×C44 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C42 |
| commutator | C4 | C2×C43 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#376 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1254 conjugacy classes of subgroups; only the 770 normal subgroups are shown.
This group has 1254 conjugacy classes of subgroups — too many to draw. Only its 770 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15,18,28)(8,12)(9,13)(10,14)(11,17,23,31)(16,27,29,35)(19,24)(20,25)(21,26)(22,30,33,36)(32,34);(5,7)(6,11,14,23)(8,16)(9,17)(10,18)(12,22,24,33)(13,15,26,28)(19,29)(20,30)(21,31)(25,27,34,35)(32,36);(1,2,3,4)(8,19)(12,24)(16,29)(20,32)(22,33)(25,34)(27,35)(30,36);(5,8,9,20,10,19,21,32)(6,12,13,25,14,24,26,34)(7,16,17,30,18,29,31,36)(11,22,15,27,23,33,28,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=e,e^2=i,b^a=be,g^a=gi,f^b=fi,d^c=di164 conjugacy classes — view the full character table.
| maximal subgroup of | #72224690, #72224701, #72224703, … |
| maximal normal subgroup of | #72224690, #72224701, #72224703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260369);