Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #347042 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2995 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 2995 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:
(6,7)(14,20)(21,24);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,20)(19,22)(23,24);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,17)(13,18)(15,21)(16,20)(19,23)(22,24);(1,4,5,3)(2,7,8,6)(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,21,20,24)Polycyclic presentation:
2^9|c^2=i,d^2=h,b^a=be,c^a=cf,d^a=dg,f^b=fh,g^b=gi,e^c=eh,e^d=ei59 conjugacy classes — view the full character table.
| maximal subgroup of | #72211144, #72237396, #72237402, … |
| maximal normal subgroup of | #72211144, #72237396, #72237402, … |
| maximal quotient of | #72219235, #72219238, #72225663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 254241);