Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23903 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,24)(23,26)(27,28);(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,21)(17,22)(19,25)(20,24)(23,27)(26,28);(16,23)(18,24)(19,26)(21,27);(6,10)(9,11)(13,16,17,23)(14,19,20,26)(15,21,22,27)(18,25,24,28)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284233, #72284594, #72285204, … |
| maximal normal subgroup of | #72284233, #72284594, #72285204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266956);