Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,19)(15,20,17,16);(6,9)(10,12)(13,15,18,17)(14,16,19,20);(1,2)(3,4)(13,16)(14,15)(17,19)(18,20);(2,4);(14,19)(16,20);(5,7,8,11)(6,10,9,12)(13,17,18,15)(14,16,19,20)Polycyclic presentation:
2^9|a^2=i,b^2=i,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^c=dh,e^c=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72305247, #72305286 |
| maximal normal subgroup of | #72305247, #72305286 |
| maximal quotient of | #72305811, #72305844 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7573651);