Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#1999 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(6,8)(9,10)(11,12)(13,14,18,22)(15,23,26,27)(16,21,17,20)(19,25,28,24);(1,2,3,4)(13,15,16,24,18,26,17,25)(14,19,20,23,22,28,21,27);(5,6,7,8)(15,24)(16,17)(19,23)(20,21)(25,26)(27,28);(2,4)(10,12)(15,24)(16,17)(19,23)(20,21)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=dei,e^c=ei,g^c=gi,e^d=ei,g^d=gi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329950);