Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12310 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1619 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| socle | C2 | C2×#1619 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
972 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 972 conjugacy classes of subgroups — too many to draw. Only its 152 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:
(1,2)(3,12)(4,9)(5,18)(6,22)(7,11)(8,15)(10,20)(13,28)(14,26)(16,29)(17,27)(19,23)(21,32)(24,31)(25,30);(1,2)(3,8)(4,9)(5,18)(6,22)(7,11)(10,20)(12,19)(13,23)(14,25)(15,26)(16,24)(17,27)(21,32)(28,31)(29,30);(1,3)(2,8)(4,13)(5,19)(6,14)(7,16)(9,23)(10,26)(11,24)(12,18)(15,20)(17,31)(21,30)(22,25)(27,28)(29,32);(1,4,7,6)(2,9,11,22)(3,13,16,14)(5,17,21,20)(8,23,24,25)(10,18,27,32)(12,28,29,26)(15,19,31,30)Polycyclic presentation:
2^9|d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,e^b=eg,f^b=fh,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214124, #72214134, #72214217, … |
| maximal normal subgroup of | #72214124, #72214134, #72214217, … |
| maximal quotient of | C4×#12310, C2×#488869, #72224802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396065);