Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244523 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1151 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 1151 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,6)(4,7)(5,8)(9,15)(10,12)(11,14)(13,16)(17,18)(19,29)(20,35)(21,39)(22,27)(23,28)(24,32)(25,37)(26,36)(30,46)(31,41)(33,43)(34,42)(38,45)(40,47)(44,48);(6,12)(13,15)(17,19)(18,24)(20,31)(21,33)(22,32)(23,34)(25,44)(26,41)(27,42)(28,43)(29,39)(30,36)(35,40)(37,47)(38,48)(45,46);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,16)(14,15)(17,20)(18,25)(19,30)(21,36)(22,37)(23,38)(24,40)(26,28)(27,45)(29,41)(31,32)(33,48)(34,47)(35,39)(42,44)(43,46);(1,4)(2,7)(3,9)(5,11)(6,13)(8,14)(10,16)(12,15)(17,20)(18,26)(19,31)(21,37)(22,36)(23,38)(24,41)(25,28)(27,45)(29,40)(30,32)(33,47)(34,48)(35,39)(42,46)(43,44)Polycyclic presentation:
2^9|e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,f^c=fh,g^c=gh,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151719);