Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| 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 | |
2337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,6,12)(3,9,18,31,17,30,26,39)(4,10,22,35,21,34,27,40)(5,11,25,38,23,36,24,37)(15,41,44,32,43,28,19,47)(16,29,45,55,46,56,52,59)(20,49,51,60,50,33,53,58)(42,54,63,64,62,61,48,57);(1,3,7,18,8,17,6,26)(2,9,13,31,14,30,12,39)(4,19,22,15,21,44,27,43)(5,16,25,45,23,46,24,52)(10,32,35,28,34,47,40,41)(11,29,38,55,36,56,37,59)(20,48,51,42,50,63,53,62)(33,57,58,54,49,64,60,61);(1,4)(2,10)(3,15)(5,20)(6,27)(7,22)(8,21)(9,28)(11,33)(12,40)(13,35)(14,34)(16,42)(17,43)(18,44)(19,26)(23,50)(24,53)(25,51)(29,54)(30,41)(31,47)(32,39)(36,49)(37,60)(38,58)(45,63)(46,62)(48,52)(55,64)(56,61)(57,59);(1,5)(2,11)(3,16)(4,20)(6,24)(7,25)(8,23)(9,29)(10,33)(12,37)(13,38)(14,36)(15,42)(17,46)(18,45)(19,48)(21,50)(22,51)(26,52)(27,53)(28,54)(30,56)(31,55)(32,57)(34,49)(35,58)(39,59)(40,60)(41,61)(43,62)(44,63)(47,64)Polycyclic presentation:
2^9|a^2=h,b^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,g^b=gi,e^c=ei,g^c=gi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285688, #72285720, #72286119, … |
| maximal normal subgroup of | #72285688, #72285720, #72286119, … |
| maximal quotient of | C2×#364521, #72343102 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271717);