Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3481 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6)(7,8)(9,10)(11,17)(12,18)(13,23)(14,16)(15,20)(19,22)(21,24);(1,2)(3,4)(5,6)(7,8)(10,16)(11,14)(17,20)(18,19)(21,22)(23,24);(6,8)(9,11,15,14)(10,17,20,16)(12,22,13,21)(18,19,23,24);(2,4)(6,8)(9,12,15,13)(10,18,20,23)(11,21,14,22)(16,19,17,24);(9,13,15,12)(10,19,20,24)(11,21,14,22)(16,18,17,23)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,d^c=di,f^e=fi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72282085, #72282214, #72282690, … |
| maximal normal subgroup of | #72282085, #72282214, #72282690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421495);