Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417647 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,9)(8,10)(11,12)(13,14,16,20,17,19,25,27)(15,21,23,26,24,28,22,18);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15)(14,18)(16,22)(17,24)(19,26)(20,28)(21,27)(23,25);(2,4)(6,10)(9,12);(14,19)(15,22)(16,25)(18,21)(23,24)(26,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321910 |
| maximal normal subgroup of | #72321910 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328114);