Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(7,11)(9,12)(14,18)(15,16)(21,22,23,24,26,25,27,28);(5,6,8,10)(7,12,11,9)(13,14,17,18)(15,19,16,20)(21,23,26,27)(22,28,25,24);(2,4)(5,7,8,11)(6,9,10,12)(13,15)(14,19)(16,17)(18,20)(21,24,26,28)(22,27,25,23);(13,16,17,15)(14,20,18,19)(21,25,26,22)(23,24,27,28)Polycyclic presentation:
2^9|a^2=efg,b^2=eg,c^2=gi,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376099);