Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500370 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#447 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(13,14)(15,21)(16,24)(17,19)(18,22)(20,25)(23,27)(26,28);(5,6)(7,11)(8,10)(9,12)(13,15,16,22,17,23,25,28)(14,18,20,21,19,26,24,27);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(14,19)(18,26)(20,24)(21,27);(7,12)(9,11);(6,10)(9,11)Polycyclic presentation:
2^9|b^2=f,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh98 conjugacy classes — view the full character table.
| maximal subgroup of | #72314723, #72314727, #72314749, … |
| maximal normal subgroup of | #72314723, #72314727, #72314749, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6413822);