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 | |
1968 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)(7,19)(8,18)(9,24)(10,27)(11,16)(12,14)(13,15)(17,29)(20,25)(21,23)(22,36)(26,35)(28,32)(30,34)(31,33);(5,7)(6,13)(8,20)(9,18)(10,24)(11,22)(12,30)(14,26)(15,29)(16,31)(17,34)(19,27)(21,25)(23,33)(28,35)(32,36);(2,4)(5,8,11,12)(6,14,16,18)(7,20,22,30)(9,13,26,31)(10,23,28,29)(15,24,33,35)(17,32,21,27)(19,34,36,25);(5,9,11,26)(6,15,16,33)(7,21,22,17)(8,13,12,31)(10,19,28,36)(14,24,18,35)(20,27,30,32)(23,34,29,25);(6,16)(7,22)(14,18)(15,33)(17,21)(20,30)(24,35)(27,32)Polycyclic presentation:
2^9|c^2=i,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, 7530828);