Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220210 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,4)(5,7)(6,8)(9,10)(11,16)(12,25,30,17)(13,31,33,18)(14,20)(15,19)(21,45,48,35)(22,49,51,36)(23,38)(24,37)(26,52,56,39)(27,40)(28,41)(29,54,46,42)(32,60,57,43)(34,44)(47,62)(50,66,61,63)(53,59)(55,64)(58,65)(67,69)(68,71)(70,72);(1,3)(2,6)(4,7)(5,8)(9,11,15,24)(10,16,19,37)(12,26,30,56)(13,22,33,51)(14,34,27,23)(17,39,25,52)(18,36,31,49)(20,44,40,38)(21,46,48,29)(28,55,59,72)(32,61,57,50)(35,54,45,42)(41,64,53,70)(43,66,60,63)(47,68,69,58)(62,71,67,65);(1,4,5,6)(2,3,7,8)(9,12,27,46,15,30,14,29)(10,17,40,54,19,25,20,42)(11,21,34,56,24,48,23,26)(13,28,57,68,33,59,32,58)(16,35,44,52,37,45,38,39)(18,41,60,71,31,53,43,65)(22,47,61,72,51,69,50,55)(36,62,66,70,49,67,63,64);(2,7)(3,8)(9,13)(10,18)(11,22)(12,28)(14,32)(15,33)(16,36)(17,41)(19,31)(20,43)(21,47)(23,50)(24,51)(25,53)(26,55)(27,57)(29,58)(30,59)(34,61)(35,62)(37,49)(38,63)(39,64)(40,60)(42,65)(44,66)(45,67)(46,68)(48,69)(52,70)(54,71)(56,72)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=egi,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293056);