Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | #37650 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2328 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)(14,16)(15,21)(18,25)(19,27)(22,28)(23,26);(2,3)(5,7)(6,9)(8,12)(10,11)(13,14,17,16)(15,19,23,22)(18,20,25,24)(21,28,26,27);(1,2,4,3)(7,12)(9,11)(13,15,17,23)(14,19,16,22)(18,26,25,21)(20,27,24,28);(6,10)(9,11)(13,16,17,14)(15,22,23,19)(18,24,25,20)(21,27,26,28);(1,3,4,2)(14,16)(19,22)(20,24)(27,28)Polycyclic presentation:
2^9|b^2=i,c^2=fh,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717366);