Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134692 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1807 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#41 | C4⋊C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
265 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 265 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,22,26,23,20,33,27)(19,28,31,44,32,45,30,43)(21,36,37,25,38,42,48,35)(24,39,41,29,34,46,40,47);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,29,22,30,37,39,23,32,38,47,33,31,48,46)(18,24,25,28,27,41,36,44,20,34,35,45,26,40,42,43);(3,9)(6,12)(10,15)(13,16)(17,20)(18,23)(19,24)(21,35)(22,26)(25,38)(27,33)(28,29)(30,41)(31,40)(32,34)(36,48)(37,42)(39,44)(43,46)(45,47)Polycyclic presentation:
2^9|a^2=f,b^2=d,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,f^b=fi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321061, #72321077, #72321093 |
| maximal normal subgroup of | #72321061, #72321077, #72321093 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41888);