Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2241 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,10)(8,12)(9,11);(5,7,10,6)(8,12,9,11)(13,14,17,20)(15,19,16,18);(2,4)(14,18)(15,16)(19,20);(5,8,10,9)(6,11,7,12)(13,15,17,16)(14,19,20,18);(5,9,10,8)(6,11,7,12)(14,19)(15,16)(18,20)Polycyclic presentation:
2^9|b^2=fi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72305248, #72305250, #72305280 |
| maximal normal subgroup of | #72305248, #72305250, #72305280 |
| maximal quotient of | #72305537, #72305703 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6584146);