Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C25 | C24 |
| commutator | C23 | C22×C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
3434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,8)(9,10,12,11)(13,14)(15,17)(16,19)(18,20)(22,23)(24,28);(14,15)(18,19)(21,22,25,23)(24,26,28,27);(1,2,3,4)(5,6,7,8)(9,11,12,10)(13,15,17,14)(16,19,20,18)(21,23,25,22)(24,27,28,26);(13,16,17,20)(14,18,15,19)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=i,d^2=g,b^a=be,c^a=cf,d^a=de,c^b=cg152 conjugacy classes — view the full character table.
| maximal subgroup of | #72233923, #72233925, #72239222, … |
| maximal normal subgroup of | #72233923, #72233925, #72239222, … |
| maximal quotient of | #72239210, #72239211, C2×#153867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 61063);