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 | |
1105 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 1105 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)(17,18,24,28)(19,29,33,40)(20,34,39,44)(21,31,25,32)(22,41,23,27)(26,37,45,38)(30,36,47,42)(35,46,48,43);(6,12)(13,15)(17,19)(18,25)(20,35)(21,28)(22,32)(23,31)(24,33)(26,43)(27,40)(29,41)(30,37)(34,42)(36,44)(38,47)(39,48)(45,46);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,16)(14,15)(17,20)(18,26)(19,30)(21,36)(22,37)(23,38)(24,39)(25,42)(27,44)(28,45)(29,46)(31,35)(32,48)(33,47)(34,41)(40,43);(1,4)(2,7)(3,9)(5,11)(6,13)(8,14)(10,16)(12,15)(17,21,24,25)(18,19,28,33)(20,36,39,42)(22,29,23,40)(26,30,45,47)(27,32,41,31)(34,35,44,48)(37,46,38,43)Polycyclic presentation:
2^9|a^2=h,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, 151725);