Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512866 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2196 |
| commutator, Frattini, cosocle | #127 | C24 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1086 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 1086 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:
(4,15)(5,18)(7,19)(8,16)(11,25)(20,32)(21,22)(23,30)(26,31)(27,28);(1,2)(3,7)(4,16)(5,9)(6,10)(8,15)(11,26)(12,19)(13,21)(14,22)(17,30)(18,24)(20,25)(23,29)(27,32)(28,31);(1,3)(2,7)(4,11)(5,12)(6,14)(8,20)(9,19)(10,22)(13,18)(15,25)(16,26)(17,27)(21,24)(23,31)(28,29)(30,32);(1,4,6,15)(2,8,10,16)(3,11,14,25)(5,17,18,29)(7,20,22,26)(9,23,24,30)(12,27,13,28)(19,31,21,32)Polycyclic presentation:
2^9|d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,g^a=gh,d^b=di,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 | #72279417, #72286316, #72286321, … |
| maximal normal subgroup of | #72279417, #72286316, #72286321, … |
| maximal quotient of | #72279452, #72340624 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420063);