Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12108 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1786 |
| prime radical | #60749 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
739 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 739 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) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2)(3,4,7,8)(5,6)(9,10,15,21,36,51,37,52,16,22,13,19,14,20,38,53)(11,23,27,44,26,54,35,17,29,43,25,42,28,50,39,45)(12,30,31,64,32,49,66,55,34,48,40,68,33,18,46,47)(24,57,58,41,61,56,67,70,60,72,62,69,59,71,65,63);(1,3)(2,4)(5,7)(6,8)(9,11)(10,17)(12,24)(13,35)(14,26)(15,39)(16,29)(18,41)(19,50)(20,42)(21,54)(22,45)(23,52)(25,37)(27,38)(28,36)(30,63)(31,65)(32,59)(33,61)(34,60)(40,67)(43,53)(44,51)(46,58)(47,57)(48,70)(49,69)(55,72)(56,68)(62,66)(64,71);(1,4)(2,3)(5,8)(6,7)(9,12)(10,18)(11,24)(13,31)(14,33)(15,40)(16,34)(17,41)(19,47)(20,48)(21,55)(22,49)(23,56)(25,58)(26,61)(27,62)(28,59)(29,60)(30,51)(32,36)(35,65)(37,46)(38,66)(39,67)(42,70)(43,71)(44,63)(45,69)(50,57)(52,68)(53,64)(54,72)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dg,e^c=ei,f^c=fgi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38475);