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 | |
1101 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 1101 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) = 128 |
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)(20,34)(21,27)(22,25)(23,28)(24,30)(26,35)(31,38)(32,40)(33,39)(36,42)(37,41)(43,45)(44,47)(46,48);(6,12)(13,15)(17,19)(18,24)(20,31)(21,32)(22,30)(23,33)(25,39)(26,38)(27,29)(28,40)(34,43)(35,44)(36,45)(37,46)(41,47)(42,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,16)(14,15)(18,25)(19,30)(21,22)(26,41)(27,28)(29,39)(31,44)(32,33)(34,42)(35,36)(43,47)(45,46);(1,4)(2,7)(3,9)(5,11)(6,13)(8,14)(10,16)(12,15)(17,20,23,37)(18,26,28,42)(19,31,33,46)(21,36,22,35)(24,38,40,48)(25,41,27,34)(29,43,39,47)(30,44,32,45)Polycyclic presentation:
2^9|d^2=h,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, 151724);