Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #22030 |
| prime radical | #3018 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,3,14,18,6,12,27,37,28,17,9,30)(4,19,21,38,22,42,31,46,23,10,29,34,51,15,39,43)(5,11,25,36,26,16,40,45,24,35,52,58,53,44,32,55)(20,33,49,57,50,41,56,61,48,47,54,59,64,62,60,63);(1,3,7,18,8,12,14,37,6,17,27,30,28,2,9,13)(4,19,21,38,22,42,31,46,23,10,29,34,51,15,39,43)(5,16,25,45,26,35,40,58,24,44,52,55,53,11,32,36)(20,47,49,59,50,62,56,63,48,33,54,57,64,41,60,61);(1,4)(2,10)(3,15)(5,20)(6,23)(7,29)(8,22)(9,31)(11,33)(12,19)(13,38)(14,39)(16,41)(17,42)(18,46)(21,27)(24,48)(25,54)(26,50)(28,51)(30,43)(32,56)(34,37)(35,47)(36,59)(40,60)(44,62)(45,63)(49,52)(53,64)(55,61)(57,58);(1,5)(2,11)(3,16)(4,20)(6,24)(7,25)(8,26)(9,32)(10,33)(12,35)(13,36)(14,40)(15,41)(17,44)(18,45)(19,47)(21,49)(22,50)(23,48)(27,52)(28,53)(29,54)(30,55)(31,56)(34,57)(37,58)(38,59)(39,60)(42,62)(43,61)(46,63)(51,64)Polycyclic presentation:
2^9|a^2=g,b^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fi,c^b=ceh,f^b=fi,e^c=ei,g^c=gi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72224961, C2×#496808, #72343184, … |
| maximal normal subgroup of | #72224961, C2×#496808, #72343184, … |
| maximal quotient of | C2×#496808 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404004);