Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #380391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2331 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14282 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
869 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,15,20,32)(4,21,24,41)(5,11)(6,12)(7,13)(8,14)(9,18,33,19)(10,23,36,29)(16,30)(17,43,50,58)(22,37,52,62)(25,53,54,35)(26,38)(27,39)(28,40)(31,48,59,49)(34,46)(42,51)(44,55)(45,56)(47,57)(60,63)(61,64);(1,3)(2,9,14,33)(4,16)(5,17)(6,18)(7,19)(8,20)(10,34,36,30)(11,31,40,59)(12,15,13,32)(21,42,41,56)(22,44)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(35,60,53,55)(37,61,62,57)(38,43,39,58)(52,63)(54,64);(1,4)(2,10,14,36)(3,16)(5,25)(6,29)(7,23)(8,24)(9,30,33,34)(11,37,40,62)(12,41,13,21)(15,42,32,56)(17,47)(18,51)(19,45)(20,46)(22,26)(27,52)(28,54)(31,57,59,61)(35,39,53,38)(43,55,58,60)(44,48)(49,63)(50,64);(1,5,6,26,8,28,7,27)(2,11,12,38,14,40,13,39)(3,17,18,48,20,50,19,49)(4,22,23,25,24,52,29,54)(9,31,15,43,33,59,32,58)(10,35,21,37,36,53,41,62)(16,44,45,47,46,63,51,64)(30,55,56,57,34,60,42,61)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288225);