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 | C2×#575 | 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:
(1,2,7,12,8,13,6,27)(3,14,17,9,18,33,28,32)(4,19,23,48,22,10,21,36)(5,24,25,11,26,38,29,37)(15,39,42,30,43,56,52,34)(16,31,46,57,45,58,44,40)(20,49,50,47,51,54,53,35)(41,55,62,64,63,59,61,60);(1,3,7,17,8,18,6,28)(2,9,12,33,13,32,27,14)(4,15,23,42,22,43,21,52)(5,16,25,46,26,45,29,44)(10,34,36,39,19,30,48,56)(11,31,38,57,37,58,24,40)(20,41,50,62,51,63,53,61)(35,59,49,60,47,55,54,64);(1,4)(2,10,13,19)(3,15)(5,20)(6,21)(7,23)(8,22)(9,30,32,34)(11,35,37,47)(12,36,27,48)(14,39,33,56)(16,41)(17,42)(18,43)(24,54,38,49)(25,50)(26,51)(28,52)(29,53)(31,55,58,59)(40,60,57,64)(44,61)(45,63)(46,62);(1,5)(2,11)(3,16)(4,20)(6,29)(7,25)(8,26)(9,31)(10,35)(12,38)(13,37)(14,40)(15,41)(17,46)(18,45)(19,47)(21,53)(22,51)(23,50)(24,27)(28,44)(30,55)(32,58)(33,57)(34,59)(36,49)(39,60)(42,62)(43,63)(48,54)(52,61)(56,64)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,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 | #72285582, #72285688, #72285996, … |
| maximal normal subgroup of | #72285582, #72285688, #72285996, … |
| maximal quotient of | C2×#368130, #72343028 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275326);