Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #477009 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17102 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1667 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 1667 conjugacy classes of subgroups — too many to draw. Only its 184 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:
(1,2,6,12,8,11,7,3)(4,13,16,24,17,25,22,9)(5,18,19,10,21,14,20,27)(15,23,30,29,32,26,31,28);(1,3,6,11,8,12,7,2)(4,9,16,25,17,24,22,13)(5,10,19,18,21,27,20,14)(15,28,30,26,32,29,31,23);(1,4)(2,9)(3,13)(5,15)(6,22)(7,16)(8,17)(10,26)(11,24)(12,25)(14,29)(18,28)(19,31)(20,30)(21,32)(23,27);(1,5)(2,10)(3,14)(4,15)(6,20)(7,19)(8,21)(9,23)(11,27)(12,18)(13,28)(16,31)(17,32)(22,30)(24,26)(25,29)Polycyclic presentation:
2^9|a^2=f,b^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,g^c=gi,f^d=fi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72219491, #72219494, #72285328, … |
| maximal normal subgroup of | #72219491, #72219494, #72285328, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384205);