Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251967 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1095 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 1095 conjugacy classes of subgroups — too many to draw. Only its 126 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 — 4 generators on 32 points:
(2,7)(3,11)(4,15)(8,23)(9,22)(10,13)(12,27)(14,29)(16,17)(19,25)(20,26)(21,32)(24,30)(28,31);(2,8,10,25)(3,12)(4,16)(7,19,13,23)(9,24,26,32)(11,27)(14,31)(15,17)(20,30,22,21)(28,29);(1,2)(3,13)(4,8)(5,10)(6,9)(7,11)(12,20)(14,30)(15,32)(16,25)(17,24)(18,26)(19,31)(21,28)(22,27)(23,29);(2,9)(3,11)(7,20)(8,24)(10,26)(12,27)(13,22)(14,28)(19,30)(21,23)(25,32)(29,31)Polycyclic presentation:
2^9|b^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^c=fh,f^d=fh44 conjugacy classes — view the full character table.
| maximal subgroup of | #72237867, #72237868 |
| maximal normal subgroup of | #72237867, #72237868 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 159163);