Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3059 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)(10,12)(11,17)(13,20)(15,22)(16,21)(18,23);(5,6)(7,8)(9,10)(11,15)(12,20)(13,21)(14,16)(17,23)(18,24)(19,22);(2,4)(6,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,24)(21,22);(2,4)(9,12,20,10,14,21,13,16)(11,18,24,15,17,22,19,23);(11,17)(15,23)(18,22)(19,24)Polycyclic presentation:
2^9|d^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dfg,f^a=fi,c^b=ch,d^b=df,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72279716, #72280303, #72307635, … |
| maximal normal subgroup of | #72279716, #72280303, #72307635, … |
| maximal quotient of | #72313040, #72313221, #72313327 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409309);