Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #386 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10712 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1071 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,10,6,12,21,32,7,11,22,31,23,33,45,54)(3,13,16,35,17,36,41,57,15,8,24,26,40,27,47,50)(4,9,19,29,20,30,44,53,18,28,42,51,43,52,60,63)(14,25,38,48,39,49,59,62,37,34,46,55,58,56,61,64);(1,3)(2,8,11,13)(4,14)(5,16)(6,17)(7,15)(9,25,28,34)(10,26,31,35)(12,27,33,36)(18,37)(19,38)(20,39)(21,41)(22,24)(23,40)(29,48,51,55)(30,49,52,56)(32,50,54,57)(42,46)(43,58)(44,59)(45,47)(53,62,63,64)(60,61);(1,4)(2,9)(3,14)(5,19)(6,20)(7,18)(8,25)(10,29)(11,28)(12,30)(13,34)(15,37)(16,38)(17,39)(21,44)(22,42)(23,43)(24,46)(26,48)(27,49)(31,51)(32,53)(33,52)(35,55)(36,56)(40,58)(41,59)(45,60)(47,61)(50,62)(54,63)(57,64)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,d^b=di,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72281050, #72281139, #72281150, … |
| maximal normal subgroup of | #72281050, #72281139, #72281150, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53572);