Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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 | #39883 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1840 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)(6,10)(7,8);(5,6)(7,12)(8,11)(9,10)(13,14,17,20)(15,21,22,28)(16,25,23,19)(18,26,27,24);(2,4)(5,7,9,8)(6,11,10,12)(13,15,16,24,17,22,23,26)(14,18,19,21,20,27,25,28);(5,8,9,7)(6,11,10,12)(15,22)(18,27)(21,28)(24,26);(13,15,23,26,17,22,16,24)(14,18,25,28,20,27,19,21)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256948, #72257047, #72257050, … |
| maximal normal subgroup of | #72256948, #72257047, #72257050, … |
| maximal quotient of | #72264557, #72264858 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718024);