Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,10,9)(7,12,8,11)(13,14)(15,19)(16,17)(18,20)(22,25)(27,28);(5,7,10,8)(6,11,9,12)(14,17)(18,19)(21,22,26,27,24,25,23,28);(5,8,10,7)(6,11,9,12)(13,15,16,20)(14,18,17,19)(22,27)(23,26)(25,28);(2,4)(5,9,10,6)(7,12,8,11)(15,20)(18,19)(21,23,24,26)(22,27,25,28)Polycyclic presentation:
2^9|a^2=efi,b^2=efg,c^2=ei,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319966, #72320123, #72320254 |
| maximal normal subgroup of | #72319966, #72320123, #72320254 |
| maximal quotient of | #72322085 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373758);