Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21776 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2576 conjugacy classes of subgroups; only the 394 normal subgroups are shown.
This group has 2576 conjugacy classes of subgroups — too many to draw. Only its 394 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,12,9,11,17,19)(7,13,15,20,14,10,16,18);(1,2,3,4)(5,7)(6,10)(8,16)(9,14)(11,13)(12,20)(15,17)(18,19);(10,13)(18,20);(6,11)(8,17)(10,13)(15,16)Polycyclic presentation:
2^9|a^2=g,b^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=ce,f^b=fi,g^b=gi,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216073, #72219640, #72219649, … |
| maximal normal subgroup of | #72216073, #72219640, #72219649, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262294);