Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14517 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1041 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,27)(3,14,17,9,18,33,28,32)(4,10,21,36,22,37,20,54)(5,23,25,11,26,40,29,39)(15,43,45,34,46,41,55,30)(16,47,49,42,48,31,24,38)(19,50,52,35,53,61,56,60)(44,57,64,62,63,58,51,59);(1,3)(2,9)(4,15)(5,24)(6,28)(7,17)(8,18)(10,34)(11,38)(12,33)(13,32)(14,27)(16,25)(19,51)(20,55)(21,45)(22,46)(23,31)(26,49)(29,48)(30,37)(35,59)(36,41)(39,42)(40,47)(43,54)(44,52)(50,58)(53,64)(56,63)(57,61)(60,62);(1,4,7,21,8,22,6,20)(2,10,12,36,13,37,27,54)(3,15,17,45,18,46,28,55)(5,19,25,52,26,53,29,56)(9,30,33,43,32,34,14,41)(11,35,40,61,39,60,23,50)(16,44,49,64,48,63,24,51)(31,57,38,62,47,58,42,59);(1,5)(2,11)(3,16)(4,19)(6,29)(7,25)(8,26)(9,31)(10,35)(12,40)(13,39)(14,42)(15,44)(17,49)(18,48)(20,56)(21,52)(22,53)(23,27)(24,28)(30,57)(32,47)(33,38)(34,58)(36,61)(37,60)(41,59)(43,62)(45,64)(46,63)(50,54)(51,55)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295379);