Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(14,18)(15,21)(16,24)(19,23)(22,27)(26,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,14,17,18)(15,21,23,19)(16,25,24,20)(22,27,26,28);(2,4)(13,15,18,21,17,23,14,19)(16,26,25,27,24,22,20,28);(6,10)(9,11)(13,16,17,24)(14,20,18,25)(15,22,23,26)(19,27,21,28);(2,4)(7,12)(9,11)(15,23)(16,24)(19,21)(20,25)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589818);