Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470092 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4460 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)(7,12)(8,11)(9,10)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,27)(24,28);(2,4)(5,7)(6,10)(8,9)(13,15)(14,18)(16,23)(17,21)(19,27)(20,25)(22,24)(26,28);(6,11)(7,8)(10,12)(15,21)(18,25)(22,23)(26,27);(6,10)(11,12)(13,16,17,24)(14,19,20,28)(15,22,21,23)(18,26,25,27);(6,10)(11,12)(14,20)(18,25)(19,28)(26,27)Polycyclic presentation:
2^9|d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,f^b=fi,g^b=gi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257668, #72263901, #72280782, … |
| maximal normal subgroup of | #72257668, #72263901, #72280782, … |
| maximal quotient of | #72264143, C2×#6481669, #72304890, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388865);