Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499919 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1464 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 176 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,12)(9,23)(10,14)(13,20)(15,21)(16,25)(17,24)(18,34)(19,28)(22,29)(26,33)(27,32)(30,36)(31,35);(2,4)(5,7)(6,11)(8,19)(9,18)(10,16)(12,28)(13,31)(14,25)(15,27)(17,22)(20,35)(21,32)(23,34)(24,29)(26,30)(33,36);(6,12)(8,20)(9,21)(11,24)(13,15)(14,29)(17,33)(18,32)(19,35)(22,36)(23,30)(25,28)(26,27)(31,34);(2,4)(7,16)(11,25)(17,19)(18,32)(24,28)(26,31)(27,34)(33,35)Polycyclic presentation:
2^9|f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fh,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72307009, #72307037, #72307041 |
| maximal normal subgroup of | #72307009, #72307037, #72307041 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407115);