Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 | C2×#3267 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2263 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)(7,10)(8,9)(11,12)(13,14,19,16)(15,20,25,17)(18,21,28,26)(22,23,27,24);(6,9)(10,12)(13,15)(14,20)(16,17)(18,23)(19,25)(21,27)(22,26)(24,28);(2,4)(5,7,8,11)(6,10,9,12)(13,16,19,14)(15,22,25,27)(17,24,20,23)(18,21,28,26);(13,17)(14,15)(16,25)(18,22)(19,20)(21,23)(24,26)(27,28);(15,23)(17,22)(18,28)(20,27)(21,26)(24,25)Polycyclic presentation:
2^9|a^2=i,c^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250121, #72256938, #72256990, … |
| maximal normal subgroup of | #72250121, #72256938, #72256990, … |
| maximal quotient of | #72264198, #72265198, #72308378 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482031);