Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23872 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13967 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2327 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,28)(3,14,17,9,18,33,29,32)(4,19,22,10,23,36,21,37)(5,24,25,11,27,38,26,39)(15,30,43,49,45,34,44,40)(16,31,47,56,48,57,46,41)(20,51,52,50,53,35,54,59)(42,61,63,58,64,60,62,55);(1,3,7,17,8,18,6,29)(2,9,12,33,13,32,28,14)(4,15,22,43,23,45,21,44)(5,16,25,47,27,48,26,46)(10,34,36,40,37,30,19,49)(11,31,38,56,39,57,24,41)(20,42,52,63,53,64,54,62)(35,58,59,60,51,55,50,61);(1,4)(2,10)(3,15)(5,20)(6,21)(7,22)(8,23)(9,30)(11,35)(12,36)(13,37)(14,40)(16,42)(17,43)(18,45)(19,28)(24,50)(25,52)(26,54)(27,53)(29,44)(31,55)(32,34)(33,49)(38,59)(39,51)(41,60)(46,62)(47,63)(48,64)(56,61)(57,58);(1,5)(2,11)(3,16)(4,20)(6,26)(7,25)(8,27)(9,31)(10,35)(12,38)(13,39)(14,41)(15,42)(17,47)(18,48)(19,50)(21,54)(22,52)(23,53)(24,28)(29,46)(30,55)(32,57)(33,56)(34,58)(36,59)(37,51)(40,60)(43,63)(44,62)(45,64)(49,61)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72317339 |
| maximal normal subgroup of | #72317339 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271833);