Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2666 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 2666 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14,16,20)(15,22,23,18)(17,19,25,21)(24,28,26,27);(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,23)(17,26)(19,28)(20,22)(21,27)(24,25);(14,19)(17,25)(18,27)(20,21)(22,28)(24,26);(2,4)(7,12)(9,11)(15,23)(18,22)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=e,b^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72258249, #72258259, #72258262 |
| maximal normal subgroup of | #72258249, #72258259, #72258262 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325073);