Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #503222 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #25771 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #24285 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1095 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1095 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,13,18,38)(4,19,23,49)(5,24,14,10)(6,11,26,35)(8,27,30,16)(9,31,33,22)(15,41,20,50)(17,37,45,63)(21,53,39,48)(25,56,40,34)(28,58,59,64)(29,57,60,44)(32,54,55,61)(36,52,62,43)(42,47,51,46);(1,3,14,27,7,18,5,16)(2,8,24,38,12,30,10,13)(4,20,39,42,23,15,21,51)(6,17,40,57,26,45,25,44)(9,28,55,62,33,59,32,36)(11,29,56,63,35,60,34,37)(19,46,53,41,49,47,48,50)(22,43,61,58,31,52,54,64);(1,4,6,22)(2,9,11,19)(3,15,17,43)(5,21,25,54)(7,23,26,31)(8,28,29,47)(10,32,34,48)(12,33,35,49)(13,36,37,41)(14,39,40,61)(16,42,44,64)(18,20,45,52)(24,55,56,53)(27,51,57,58)(30,59,60,46)(38,62,63,50);(1,3,14,27,7,18,5,16)(2,8,24,38,12,30,10,13)(4,15,39,51,23,20,21,42)(6,17,40,57,26,45,25,44)(9,28,55,62,33,59,32,36)(11,29,56,63,35,60,34,37)(19,47,53,50,49,46,48,41)(22,43,61,58,31,52,54,64)Polycyclic presentation:
2^9|a^2=i,b^2=ei,c^2=f,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,f^b=fh,g^c=gh,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72303765, #72303766, #72312584, … |
| maximal normal subgroup of | #72303765, #72303766, #72312584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410422);