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 | |
3425 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)(11,19)(12,16)(13,22)(14,18)(15,20)(17,23)(21,24);(1,2)(3,4)(5,6)(7,8)(9,11,14,12)(10,15,18,24)(13,21,23,20)(16,22,19,17);(6,8)(11,12)(15,24)(16,19)(20,21);(2,4)(6,8)(9,12,14,11)(10,16,18,19)(13,21,23,20)(15,22,24,17);(10,17)(13,23)(15,16)(18,22)(19,24)(20,21)Polycyclic presentation:
2^9|b^2=i,d^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 | #72257523, #72257588, #72263868, … |
| maximal normal subgroup of | #72257523, #72257588, #72263868, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421498);