Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14015 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1545 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)(3,9,18,31,17,30,27,40)(4,10,23,36,21,34,22,35)(5,11,26,39,24,37,25,38)(15,41,19,46,43,28,48,32)(16,29,44,55,45,56,53,60)(20,49,52,58,50,33,51,59)(42,54,47,57,62,61,64,63);(1,3,7,18,8,17,6,27)(2,9,13,31,14,30,12,40)(4,19,23,43,21,48,22,15)(5,16,26,44,24,45,25,53)(10,32,36,41,34,46,35,28)(11,29,39,55,37,56,38,60)(20,47,52,62,50,64,51,42)(33,57,59,61,49,63,58,54);(1,4,6,22,8,21,7,23)(2,10,12,35,14,34,13,36)(3,15,18,19,17,43,27,48)(5,20,25,51,24,50,26,52)(9,28,31,32,30,41,40,46)(11,33,38,58,37,49,39,59)(16,42,44,47,45,62,53,64)(29,54,55,57,56,61,60,63);(1,5)(2,11)(3,16)(4,20)(6,25)(7,26)(8,24)(9,29)(10,33)(12,38)(13,39)(14,37)(15,42)(17,45)(18,44)(19,47)(21,50)(22,51)(23,52)(27,53)(28,54)(30,56)(31,55)(32,57)(34,49)(35,58)(36,59)(40,60)(41,61)(43,62)(46,63)(48,64)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,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,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285688, #72285720, #72286143, … |
| maximal normal subgroup of | #72285688, #72285720, #72286143, … |
| maximal quotient of | C2×#368085, #72343017 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275281);