Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622820 |
| 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 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,12)(8,14)(9,22)(10,26)(11,15)(13,18)(16,24)(17,27)(19,28)(20,30)(21,33)(23,29)(25,36)(31,34)(32,35);(1,2,3,4)(5,7)(6,12)(8,20)(9,18)(10,17)(11,19)(13,22)(14,30)(15,28)(16,29)(21,35)(23,24)(25,34)(26,27)(31,36)(32,33);(5,7)(6,13)(8,17)(9,23)(10,20)(11,19)(12,22)(14,29)(15,31)(16,30)(18,24)(21,34)(25,35)(26,32)(27,33)(28,36);(5,8,11,10)(6,14,15,26)(7,17,19,20)(9,21,25,24)(12,27,28,30)(13,29,31,32)(16,22,33,36)(18,23,34,35);(6,15)(12,28)(13,31)(14,26)(16,33)(22,36)(27,30)(29,32)Polycyclic presentation:
2^9|b^2=f,d^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=di,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72320324, #72320347, #72320537, … |
| maximal normal subgroup of | #72320324, #72320347, #72320537, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530133);