Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7394 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1343 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | #379 | C23 |
| socle | C2 | C2×#1343 |
| prime radical | #23443 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ #2280 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
724 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 724 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,8)(5,17)(6,10)(7,13,20,27)(9,18)(12,24)(14,23)(15,28)(16,22)(19,25,30,32)(21,29)(26,31);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,23)(10,20)(11,17)(15,25)(16,26)(18,27)(21,31)(22,30)(24,28)(29,32);(1,4,6,16)(2,8,10,22)(3,12,14,26)(5,15,18,29)(7,19,20,30)(9,21,17,28)(11,24,23,31)(13,25,27,32)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=g,b^a=bd,d^a=df,f^a=fh,g^a=gi,d^b=dg,e^b=efg,h^b=hi,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72338379, #72338381, #72338382, … |
| maximal normal subgroup of | #72338379, #72338381, #72338382, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57363);