Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124660 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1510 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C2 | #7606 |
| prime radical | C2×#575 | C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
495 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 495 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,11,13,28,15,27,14,23)(4,16,18,8)(5,10)(6,9)(7,12,21,17,22,29,26,30)(19,24)(20,31,32,25);(1,3)(2,7)(4,17)(5,13)(6,15)(8,23)(9,22)(10,26)(11,25)(12,20)(14,19)(16,28)(18,30)(21,24)(27,31)(29,32);(4,18)(8,16)(11,27)(12,29)(17,30)(20,32)(23,28)(25,31)Polycyclic presentation:
2^9|a^2=e,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=di,e^d=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219607, #72226637, #72226639 |
| maximal normal subgroup of | #72219607, #72226637, #72226639 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31856);