Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395154 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×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 | |
1990 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)(9,10)(11,12)(13,14,18,21)(15,22,25,28)(16,20,26,27)(17,24,19,23);(5,6)(7,8)(13,15,17,16,18,25,19,26)(14,20,23,22,21,27,24,28);(1,2,3,4)(6,8)(14,21)(20,27)(22,28)(23,24);(10,12)(13,16,17,25,18,26,19,15)(14,22,23,27,21,28,24,20)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=fi,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264180, #72264237, #72264342, … |
| maximal normal subgroup of | #72264180, #72264237, #72264342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305619);