Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5513 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,13,12)(11,15,16,14)(17,18)(19,23)(20,24)(21,22);(6,8)(10,12)(14,15)(17,19)(18,20)(21,24)(22,23);(5,6,7,8)(11,16)(14,15)(18,22)(19,24);(1,2)(3,4)(9,11)(10,14)(12,15)(13,16)(17,19)(18,23)(20,22)(21,24);(9,12,13,10)(11,15,16,14)(17,20,21,23)(18,19,22,24)Polycyclic presentation:
2^9|a^2=fg,c^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232074, #72232083, #72232109, … |
| maximal normal subgroup of | #72232074, #72232083, #72232109, … |
| maximal quotient of | #72278224, #72279113, C2×#1084838, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 992034);