Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470137 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,19,15)(16,24,26,28)(17,21,18,23)(20,27,22,25);(1,2,3,4)(7,8)(10,11)(13,15,19,14)(16,20,26,22)(17,21,18,23)(24,27,28,25);(5,7,9,8)(6,10,12,11)(13,16)(14,20)(15,22)(17,25)(18,27)(19,26)(21,28)(23,24);(5,8,9,7)(6,11,12,10)(13,17,19,18)(14,21,15,23)(16,25,26,27)(20,28,22,24);(6,12)(10,11)(14,15)(20,22)(21,23)(24,28)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=hi,d^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394900);