Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438955 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(6,8)(9,10,13,18,14,19,21,15)(11,16,20,24,12,17,22,23);(2,4)(10,15)(11,20)(12,22)(13,21)(18,19)(23,24);(5,6,7,8)(9,11,13,20,14,12,21,22)(10,16,18,24,19,17,15,23);(9,12,13,22,14,11,21,20)(10,17,18,23,19,16,15,24);(11,12)(16,17)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72279716, #72279794, #72280255, … |
| maximal normal subgroup of | #72279716, #72279794, #72280255, … |
| maximal quotient of | #72312948, #72313040 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372854);