Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1957 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,21)(9,24)(10,28)(11,14)(12,13)(15,26)(16,25)(18,35)(19,34)(20,22)(23,33)(27,36)(29,32)(30,31);(5,7)(6,12)(8,16)(9,21)(10,29)(11,19)(13,26)(14,30)(15,32)(17,24)(18,22)(20,25)(23,28)(27,33)(31,35)(34,36);(2,4);(5,8,11,23)(6,13,14,31)(7,18,19,15)(9,17,27,34)(10,30,25,12)(16,33,28,21)(20,24,29,36)(22,35,32,26);(6,14)(7,19)(13,31)(15,18)(16,28)(20,29)(21,33)(24,36)Polycyclic presentation:
2^9|d^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fh,h^a=hi,d^b=df,e^b=ei,f^b=fh,h^b=hi,f^d=fh,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72320537 |
| maximal normal subgroup of | #72320537 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530826);