Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23);(9,11,13,21,14,19,12,20)(10,15,16,18,17,23,22,24);(2,4)(11,19)(15,23)(18,24)(20,21);(6,8);(9,12,14,13)(10,16,17,22)(11,20,19,21)(15,18,23,24)Polycyclic presentation:
2^9|b^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ci104 conjugacy classes — view the full character table.
| maximal subgroup of | #72306123, #72306125, #72306151 |
| maximal normal subgroup of | #72306123, #72306125, #72306151 |
| maximal quotient of | #72306446, #72306478, #72308727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536194);