Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,11,16,25)(8,18,20,28)(9,14)(10,13)(12,19,26,21)(15,29)(17,24,32,36)(22,27)(23,33)(30,35)(31,34);(1,2)(3,4)(5,7)(6,11,13,25)(8,15)(9,17)(10,16)(12,23,26,34)(14,24,27,36)(18,29,28,35)(19,31)(20,30)(21,33)(22,32);(2,4)(5,8)(6,12)(7,15)(9,21)(10,20)(11,23)(13,26)(14,28)(16,30)(17,33)(18,27)(19,22)(24,35)(25,34)(29,36)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,b^a=bd,c^a=ce,d^a=di,h^a=hi,c^b=cf,d^b=di,g^b=gi,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239324, #72239349, #72239427, … |
| maximal normal subgroup of | #72239324, #72239349, #72239427, … |
| maximal quotient of | #72267250, #72279580, #72281543, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30794);