Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #265257 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1237 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 1237 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,22)(18,21)(19,23)(20,24);(1,2,6,8,4,7,3,5)(9,11)(10,14)(12,17)(13,18)(15,22)(16,21)(19,24)(20,23);(2,7)(5,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,23)(18,24)(21,22);(1,2,6,8,4,7,3,5)(12,19)(15,23)(17,24)(20,22)Polycyclic presentation:
2^9|b^2=eh,d^2=eh,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,d^c=dhi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72306408, #72306432, #72306495, … |
| maximal normal subgroup of | #72306408, #72306432, #72306495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 172456);