Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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×#3284 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2167 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)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,19,24,27)(16,21,26,23)(18,22,28,25);(6,9)(10,12)(13,15)(14,19)(16,25)(17,24)(18,21)(20,27)(22,26)(23,28);(2,4)(5,7,8,11)(6,10,9,12)(14,20)(15,21)(16,19)(18,28)(23,24)(26,27);(13,16,17,26)(14,21,20,23)(15,22,24,25)(18,19,28,27);(15,23)(16,19)(18,28)(21,24)(22,25)(26,27)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,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 | #72307127, #72307174, #72307563 |
| maximal normal subgroup of | #72307127, #72307174, #72307563 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482064);