Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| 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 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12);(5,7,8,12)(6,9,10,11)(13,14,17,20)(15,19,16,18);(2,4)(14,18)(15,16)(19,20);(6,10)(9,11);(7,12)(9,11)(13,15)(16,17)(18,19)Polycyclic presentation:
2^9|b^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283250, #72283368, #72283651, … |
| maximal normal subgroup of | #72283250, #72283368, #72283651, … |
| maximal quotient of | #72304991, #72305665, #72305799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6575334);