Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151634 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12424 |
| commutator, Frattini, cosocle | C4×C16 | C23 |
| LCS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
176 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 44 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:
(2,5)(3,10)(4,7)(6,8)(11,21)(12,13)(14,22)(15,27)(16,28)(17,26)(18,19)(20,23)(24,25)(29,32)(30,31);(1,2,7,12,6,11,19,22,9,14,18,21,8,13,4,5)(3,10,23,27,16,25,29,31,17,26,30,32,24,28,15,20);(1,3)(2,10)(4,15)(5,20)(6,16)(7,23)(8,24)(9,17)(11,25)(12,27)(13,28)(14,26)(18,30)(19,29)(21,32)(22,31)Polycyclic presentation:
2^9|b^2=df,d^2=fi,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fi,g^a=gh,h^a=hi,e^c=eg,g^c=gh,h^c=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72214100, #72214187, #72214214, … |
| maximal normal subgroup of | #72214100, #72214187, #72214214, … |
| maximal quotient of | C2×#151634, #72216270, #72225437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58830);