Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
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 | |
6406 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)(11,16)(12,17)(13,15)(14,18)(19,24)(20,23)(21,22);(5,6)(7,8)(10,15)(11,19)(12,13)(16,23)(17,18)(22,24);(2,4)(6,8)(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,23)(18,24);(6,8)(9,12,14,13)(10,17,18,15)(11,20,19,21)(16,23,24,22);(11,19)(16,24)(20,21)(22,23)Polycyclic presentation:
2^9|d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72305329, #72305339, #72305342 |
| maximal normal subgroup of | #72305329, #72305339, #72305342 |
| maximal quotient of | #72305900, #72306008, #72306038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344618);