Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62253 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,25)(23,28)(24,27);(1,3,4,2)(6,9)(7,11)(15,23)(19,27)(22,28)(24,25);(13,15,17,25,18,23,16,24)(14,19,20,22,21,27,26,28);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,20,21,26)(15,24,23,25)(19,22,27,28);(6,9)(10,12)(13,16,18,17)(14,20,21,26)(15,24,23,25)(19,22,27,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df104 conjugacy classes — view the full character table.
| maximal subgroup of | #72311746, #72311888, #72311937, … |
| maximal normal subgroup of | #72311746, #72311888, #72311937, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6296503);