Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393934 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#2021 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(6,7)(9,10)(11,15)(12,16)(13,14)(17,23)(18,20)(19,21)(22,24);(1,2)(3,4)(5,6,8,7)(10,14)(15,20)(16,21)(22,23);(2,4)(9,11)(10,15)(12,17)(13,18)(14,20)(16,22)(19,24)(21,23);(5,7,8,6)(9,12,13,19)(10,16,14,21)(11,17,18,24)(15,22,20,23)Polycyclic presentation:
2^9|b^2=ei,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,h^c=hi,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72251366, #72251390, #72251395 |
| maximal normal subgroup of | #72251366, #72251390, #72251395 |
| maximal quotient of | #72251440 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302337);