Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4960 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)(9,10,13,16,14,17,12,21)(11,15,20,23,18,22,19,24);(5,6)(7,8)(9,10)(11,15)(12,16)(13,21)(14,17)(18,22)(19,23)(20,24);(2,4)(6,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24);(6,8)(11,18)(15,22)(19,20)(23,24);(9,12,14,13)(10,16,17,21)(11,19,18,20)(15,23,22,24)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239649, #72249710, #72281405, … |
| maximal normal subgroup of | #72239649, #72249710, #72281405, … |
| maximal quotient of | #72239726, #72305542, C2×#6437756 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344952);