Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,7,8)(11,19,22,21)(16,23,24,20);(1,2)(3,4)(6,8)(9,10,13,12,15,18,14,17)(11,20,19,16,22,23,21,24);(2,4)(6,8)(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,23)(18,24);(6,8)(9,12,13,18,15,17,14,10)(11,20,21,24,22,23,19,16)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=fh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72309747, #72310134, #72310141 |
| maximal normal subgroup of | #72309747, #72310134, #72310141 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285100);