Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34743 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2847 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,15,18,14,17,19,24)(11,13,12,21,20,23,22,16);(1,2,3,4)(9,11)(10,13)(12,15)(14,20)(16,24)(17,23)(18,21)(19,22);(6,8)(9,12)(10,16)(11,19)(13,18)(14,22)(15,20)(17,21)(23,24);(9,13,15,21,14,23,19,16)(10,12,18,20,17,22,24,11);(11,20)(12,22)(13,23)(16,21)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=h,h^2=i,b^a=bf,c^a=cg,c^b=ci,e^b=ei,e^c=ei,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72307642, #72307670, #72307680, … |
| maximal normal subgroup of | #72307642, #72307670, #72307680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279976);