Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682420 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×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 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,8,9)(7,11,12,10)(14,17)(15,18)(19,20);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,17)(15,19,20,18);(2,4)(6,9)(10,11)(13,15,17,19,16,20,14,18);(5,7,8,12)(6,10,9,11)(13,14,16,17)(15,19,20,18);(2,4)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307734, #72307778, #72308015 |
| maximal normal subgroup of | #72307734, #72307778, #72308015 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592040);