Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394108 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14734 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1829 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,13,8,14)(3,15,18,32,20,33,19,9)(4,21,23,39,24,10,29,36)(5,25,26,11,28,37,27,38)(16,40,43,34,44,57,49,30)(17,45,46,41,47,31,48,55)(22,51,52,35,53,59,54,50)(42,56,61,64,62,58,63,60);(1,3,8,18,7,20,6,19)(2,9,14,15,13,32,12,33)(4,16,29,43,24,44,23,49)(5,17,26,48,28,47,27,46)(10,34,39,57,21,30,36,40)(11,31,37,41,38,45,25,55)(22,42,52,63,53,62,54,61)(35,58,59,64,50,56,51,60);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,29)(9,30)(11,35)(12,36)(13,21)(14,39)(15,40)(17,42)(18,43)(19,49)(20,44)(25,51)(26,52)(27,54)(28,53)(31,56)(32,34)(33,57)(37,59)(38,50)(41,60)(45,58)(46,61)(47,62)(48,63)(55,64);(1,5)(2,11)(3,17)(4,22)(6,27)(7,28)(8,26)(9,31)(10,35)(12,25)(13,38)(14,37)(15,41)(16,42)(18,48)(19,46)(20,47)(21,50)(23,54)(24,53)(29,52)(30,56)(32,45)(33,55)(34,58)(36,51)(39,59)(40,60)(43,63)(44,62)(49,61)(57,64)Polycyclic presentation:
2^9|a^2=e,b^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285493, #72285536, C2×#396535, … |
| maximal normal subgroup of | #72285493, #72285536, C2×#396535, … |
| maximal quotient of | C2×#396535 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303731);