Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153189 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13348 |
| commutator, Frattini, cosocle | C2×#127 | C23 |
| prime radical | C2×#453 | C22 |
| LCS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
528 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 50 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)(3,7)(4,15,18,8)(5,10)(6,9)(11,19)(12,27,30,20)(13,22)(14,21)(16,25,17,26)(23,31)(24,28,32,29);(1,3,6,14)(2,7,9,21)(4,12,18,30)(5,13,19,31)(8,24,15,32)(10,22,11,23)(16,29,17,28)(20,25,27,26);(1,4)(2,8)(3,12)(5,16)(6,18)(7,20)(9,15)(10,26)(11,25)(13,28)(14,30)(17,19)(21,27)(22,24)(23,32)(29,31)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fgh,g^a=gi,c^b=cf,d^b=dh,g^b=gh,d^c=dg,g^c=gi,h^c=hi,f^d=fh,g^f=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72231244, C2×#153190, #72279325, … |
| maximal normal subgroup of | #72231244, C2×#153190, #72279325, … |
| maximal quotient of | C2×#153190, #72279333, #72279337 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60386);