Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34633 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,12,17,13,16,21,22)(11,15,19,23,18,24,20,14);(5,6)(7,8)(9,11,13,18)(10,14,16,23)(12,20,21,19)(15,17,24,22);(1,2,3,4)(6,8)(9,11)(10,15)(12,19)(13,18)(14,22)(16,24)(17,23)(20,21);(6,8)(11,18)(14,23)(15,24)(19,20);(10,16)(14,23)(15,24)(17,22)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72342365, #72342367, #72342379 |
| maximal normal subgroup of | #72342365, #72342367, #72342379 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350186);