Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #474304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16936 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,28,7,13,6,12)(3,14,17,32,18,43,29,9)(4,19,26,35,22,10,21,39)(5,11,27,40,25,38,24,37)(15,44,46,33,47,41,52,30)(16,48,50,60,51,42,53,31)(20,54,56,36,55,34,23,57)(45,59,49,61,63,62,64,58);(1,3,8,29,7,18,6,17)(2,9,12,14,13,32,28,43)(4,15,26,52,22,47,21,46)(5,16,24,50,25,51,27,53)(10,33,35,41,19,30,39,44)(11,31,40,42,38,60,37,48)(20,45,23,49,55,63,56,64)(34,61,57,59,54,58,36,62);(1,4,7,22)(2,10,13,19)(3,15,18,47)(5,23,25,56)(6,26,8,21)(9,30,32,33)(11,36,38,57)(12,39,28,35)(14,41,43,44)(16,49,51,64)(17,52,29,46)(20,27,55,24)(31,59,60,62)(34,40,54,37)(42,61,48,58)(45,53,63,50);(1,5)(2,11,13,38)(3,16)(4,20)(6,27)(7,25)(8,24)(9,31,32,60)(10,34,19,54)(12,40,28,37)(14,42,43,48)(15,45)(17,53)(18,51)(21,56)(22,55)(23,26)(29,50)(30,58,33,61)(35,57,39,36)(41,59,44,62)(46,64)(47,63)(49,52)Polycyclic presentation:
2^9|a^2=efg,b^2=efgh,c^2=i,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381501);