Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(5,6,8,7)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24);(1,2,3,4);(5,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23);(11,18)(14,17)(19,24)(21,23);(10,15)(14,17)(16,22)(21,23);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,d^c=dh176 conjugacy classes — view the full character table.
| maximal subgroup of | #72240819, #72250400, #72250438, … |
| maximal normal subgroup of | #72240819, #72250400, #72250438, … |
| maximal quotient of | #72240753, #72240757, #72279850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532727);