Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 50331648 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
18893 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,8)(10,12)(15,21)(16,23)(19,24)(20,25);(5,6)(7,8)(14,18)(19,24)(20,25)(26,28);(9,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)Polycyclic presentation:
2^9|d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
| maximal subgroup of | #72261788, #72334941 |
| maximal normal subgroup of | #72261788, #72334941 |
| maximal quotient of | #72334952, #72334955, #72334958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215241);