Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,8)(10,11)(13,14,16,20,17,19,25,27)(15,21,23,26,22,28,24,18);(1,2,3,4)(5,6,9,8)(7,11,12,10)(15,22)(18,26)(21,28)(23,24);(2,4)(5,7,9,12)(6,10,8,11)(13,15)(14,18)(16,24)(17,22)(19,26)(20,28)(21,27)(23,25);(5,8,9,6)(7,10,12,11)(14,19)(16,25)(18,26)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=e,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,f^c=fi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372060);