Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512884 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2196 |
| commutator, Frattini, cosocle | #124 | C24 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1054 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1054 conjugacy classes of subgroups — too many to draw. Only its 112 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:
(2,8)(4,5)(6,20)(9,24)(10,12)(11,23)(13,18)(14,31)(22,26)(25,27)(28,32)(29,30);(1,2,7,12)(3,9,16,28)(4,8,5,10)(6,11,20,30)(13,22,18,25)(14,32,31,24)(15,27,21,26)(17,23,19,29);(1,3)(2,9)(4,13)(5,18)(6,21)(7,16)(8,22)(10,25)(11,26)(12,28)(14,19)(15,20)(17,31)(23,24)(27,30)(29,32);(1,4,7,5)(2,10,12,8)(3,13,16,18)(6,17,20,19)(9,25,28,22)(11,29,30,23)(14,21,31,15)(24,26,32,27)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=gh,d^b=di,e^b=ei,f^b=fghi,h^b=hi,e^c=eg,g^c=gi,h^c=hi,f^e=fhi,g^e=gi,g^f=gi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72279638, #72286053 |
| maximal normal subgroup of | #72279638, #72286053 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420083);