Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573477 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24354 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14,16,20)(15,19,23,18)(17,21,25,28)(22,27,24,26);(6,9)(10,12)(13,15)(14,18)(16,23)(17,22)(19,20)(21,26)(24,25)(27,28);(2,4)(15,22)(17,25)(18,26)(19,27)(21,28)(23,24);(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,23)(17,22)(18,20)(21,27)(24,25)(26,28);(14,20)(15,22)(17,25)(18,27)(19,26)(23,24)Polycyclic presentation:
2^9|a^2=f,d^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72256938, #72256947, #72256992, … |
| maximal normal subgroup of | #72256938, #72256947, #72256992, … |
| maximal quotient of | #72264374, #72264782, #72265189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483122);