Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,23)(20,24);(2,4)(5,6)(7,8);(6,8)(9,11,13,21,14,20,12,19)(10,15,16,18,17,23,22,24);(6,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,18,23,24);(11,20)(15,23)(18,24)(19,21)Polycyclic presentation:
2^9|c^2=g,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72249507, #72249525, #72249565, … |
| maximal normal subgroup of | #72249507, #72249525, #72249565, … |
| maximal quotient of | #72264155, #72264369, #72264372, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349966);