Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500807 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24260 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1017 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)(3,11)(4,10)(5,24)(6,27)(7,29)(8,14)(9,16)(12,25)(13,32)(15,36)(17,49)(18,41)(19,52)(20,54)(21,57)(22,38)(23,39)(26,34)(28,42)(30,40)(31,44)(33,46)(35,53)(37,55)(43,48)(45,64)(47,62)(50,58)(51,63)(56,61)(59,60);(1,3,6,16,8,18,25,5)(2,9,12,11,14,24,27,41)(4,19,21,53,23,51,55,56)(7,30,28,26,32,17,50,48)(10,35,37,52,39,61,57,63)(13,43,42,49,29,34,58,40)(15,45,46,31,47,59,20,22)(33,60,36,44,54,64,62,38);(1,4)(2,10)(3,15)(5,20)(6,21)(7,31)(8,23)(9,33)(11,36)(12,37)(13,44)(14,39)(16,46)(17,51)(18,47)(19,30)(22,32)(24,54)(25,55)(26,53)(27,57)(28,59)(29,38)(34,61)(35,43)(40,63)(41,62)(42,64)(45,50)(48,56)(49,52)(58,60);(1,5,25,18,8,16,6,3)(2,11,27,9,14,41,12,24)(4,20,55,47,23,46,21,15)(7,26,50,30,32,48,28,17)(10,36,57,33,39,62,37,54)(13,40,58,34,29,49,42,43)(19,22,56,59,51,31,53,45)(35,44,63,60,61,38,52,64)Polycyclic presentation:
2^9|b^2=e,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72312282, #72312338, #72312376, … |
| maximal normal subgroup of | #72312282, #72312338, #72312376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408005);