Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417508 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #35390 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,24)(18,22)(19,21)(20,23);(1,2,3,4)(9,11)(10,14)(12,19)(13,18)(15,21)(16,22)(17,20)(23,24);(9,12,13,20)(10,15,16,23)(11,17)(14,21)(18,19)(22,24);(2,4)(6,8)Polycyclic presentation:
2^9|b^2=e,c^2=fi,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72251366, #72251399, #72251401 |
| maximal normal subgroup of | #72251366, #72251399, #72251401 |
| maximal quotient of | #72251474 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329994);