Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5284 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)(10,16)(11,17)(13,14)(18,19)(21,22)(23,24);(1,2,3,4)(5,6)(7,8)(9,10,13,16,15,19,14,18)(11,20,22,24,21,12,17,23);(6,8)(9,11,13,22,15,21,14,17)(10,12,16,23,19,20,18,24);(6,8)(9,12)(10,17)(11,16)(13,23)(14,24)(15,20)(18,21)(19,22);(11,21)(12,20)(17,22)(23,24)Polycyclic presentation:
2^9|b^2=f,c^2=gi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249581, #72249632, #72249641, … |
| maximal normal subgroup of | #72249581, #72249632, #72249641, … |
| maximal quotient of | #72264583, #72264592, #72307703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350465);