Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346446 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2641 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 2641 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,8)(10,15)(11,13)(16,23)(17,20)(18,21)(22,24);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,20)(14,17)(18,22)(19,24)(21,23);(1,3,4,7)(2,5,6,8)(9,11,14,13)(10,15,17,20)(12,18,19,21)(16,22,24,23);(9,12,14,19)(10,16,17,24)(11,18,13,21)(15,22,20,23)Polycyclic presentation:
2^9|c^2=hi,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72235013, #72235017, #72235018, … |
| maximal normal subgroup of | #72235013, #72235017, #72235018, … |
| maximal quotient of | #72261622 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253647);