Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3484 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5370 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)(6,8)(10,15)(12,13)(16,19)(17,18)(21,22)(23,24);(2,4)(9,10,14,18)(11,19,20,24)(12,17,13,15)(16,21,23,22);(5,6,7,8)(9,11)(10,16)(12,22)(13,21)(14,20)(15,19)(17,24)(18,23);(10,15)(12,13)(16,19)(17,18)(21,22)(23,24);(11,20)(16,23)(19,24)(21,22)Polycyclic presentation:
2^9|b^2=i,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72249598, #72249615, #72249651, … |
| maximal normal subgroup of | #72249598, #72249615, #72249651, … |
| maximal quotient of | #72264796, #72264834, #72308693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349874);