Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C24 |
| commutator, socle | C23 | C23×C8 |
| prime radical | C22×D4 | C2×C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
641 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)(7,10)(8,9)(11,12)(13,14,17,21,18,15,22,24,16,19,25,27,26,23,20,28);(6,9)(10,12);(2,4)(5,7,8,11)(6,10,9,12)(14,19)(15,23)(21,27)(24,28);(13,15,17,24,18,19,22,27,16,23,25,28,26,14,20,21)Polycyclic presentation:
2^9|a^2=g,c^2=e,d^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,d^c=di152 conjugacy classes — view the full character table.
| maximal subgroup of | #72319372, #72319574, #72319580, … |
| maximal normal subgroup of | #72319372, #72319574, #72319580, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403101);