Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | |
3306 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)(10,15)(12,20)(13,21)(16,22)(18,24)(19,23);(1,2)(3,4)(5,6)(7,8)(9,10,14,12)(11,16,17,18)(13,20,21,15)(19,24,23,22);(6,8)(9,11)(10,16)(12,18)(13,19)(14,17)(15,22)(20,24)(21,23);(2,4)(6,8)(11,17)(16,18)(19,23)(22,24);(9,12,14,10)(11,18,17,16)(13,15,21,20)(19,22,23,24)Polycyclic presentation:
2^9|b^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,f^b=fi,d^c=di,f^e=fi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72281826, #72282085, #72282154, … |
| maximal normal subgroup of | #72281826, #72282085, #72282154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421228);