Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,19)(16,18)(17,20);(5,7)(6,10)(8,11)(9,12)(13,15)(14,17)(16,20)(18,19)(21,22)(23,26)(24,25)(27,28);(14,18)(17,19)(22,25)(26,28);(6,11)(7,12)(15,20)(17,19)(21,23,24,27)(22,26,25,28);(5,8,9,10)(6,7,11,12)Polycyclic presentation:
2^9|a^2=i,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df128 conjugacy classes — view the full character table.
| maximal subgroup of | #72304116, #72304117, #72304126, … |
| maximal normal subgroup of | #72304116, #72304117, #72304126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 428052);