Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #35348 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,18)(19,23)(20,22)(21,24);(10,14)(13,16)(17,19)(18,21)(20,24)(22,23);(19,24)(21,23);(18,22)(21,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|b^2=h,f^2=g,b^a=bg,c^a=ch,c^b=ci,e^b=ei,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72241235, #72241236, #72241249, … |
| maximal normal subgroup of | #72241235, #72241236, #72241249, … |
| maximal quotient of | C2×#7628732, #72342172 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535928);