Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622976 |
| 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 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1954 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)(7,16)(8,13)(9,24)(10,22)(11,15)(12,20)(14,17)(18,29)(19,30)(21,35)(23,28)(25,36)(26,33)(27,34)(31,32);(5,7,17,32,10,21,25,34,11,19,23,33,8,18,9,20)(6,12,24,29,13,26,28,30,15,27,36,35,22,31,14,16);(2,4)(5,6)(7,12)(8,22)(9,14)(10,13)(11,15)(16,20)(17,24)(18,31)(19,27)(21,26)(23,36)(25,28)(29,32)(30,34)(33,35);(5,8,11,10)(6,13,15,22)(7,18,19,21)(9,23,25,17)(12,26,27,31)(14,24,28,36)(16,29,30,35)(20,33,34,32);(7,19)(12,27)(16,30)(18,21)(20,34)(26,31)(29,35)(32,33)Polycyclic presentation:
2^9|b^2=fh,d^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,f^a=fh,h^a=hi,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72320284, #72320296, #72320537, … |
| maximal normal subgroup of | #72320284, #72320296, #72320537, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530174);