Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×C4×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,14,10,13,22,26)(7,11,17,25,16,24,20,29)(8,12,19,27,18,28,21,30)(15,31,33,36,32,23,34,35);(2,4)(5,7)(6,11,13,24)(8,15)(9,20)(10,16)(12,23,28,31)(14,29,26,25)(17,22)(18,32)(19,34)(21,33)(27,36,30,35);(5,8,9,21,10,18,22,19)(6,12,26,27,13,28,14,30)(7,15,20,33,16,32,17,34)(11,23,25,36,24,31,29,35);(7,16)(11,24)(15,32)(17,20)(23,31)(25,29)(33,34)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72251179, #72251193, #72251209, … |
| maximal normal subgroup of | #72251179, #72251193, #72251209, … |
| maximal quotient of | #72258168, C2×#354057, #72314706 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261253);