Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1279 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,19)(12,16)(13,18)(14,17)(15,21)(20,24)(22,23)(25,26)(27,36)(28,40)(29,34)(30,45)(31,38)(32,43)(33,42)(35,44)(37,48)(39,49)(41,52)(46,51)(47,56)(50,54)(53,55);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24)(25,27)(26,31)(28,41)(29,38)(30,39)(32,50)(33,52)(34,36)(35,49)(37,42)(40,48)(43,56)(44,55)(45,53)(46,54)(47,51);(2,5)(6,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28)(26,32)(27,37)(29,42)(30,46)(31,47)(33,35)(34,51)(36,50)(38,41)(39,56)(40,45)(43,44)(48,49)(52,53)(54,55);(1,3,4,7)(2,6,5,8)(10,17)(15,19)(16,18)(23,24)(25,28)(26,33)(27,37)(29,42)(30,43)(31,48)(32,35)(34,40)(36,52)(38,41)(39,54)(44,46)(45,51)(47,49)(50,53)(55,56);(2,5)(6,8)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(27,38)(28,42)(31,36)(32,51)(33,40)(39,55)(43,46)(49,53)Polycyclic presentation:
2^9|c^2=fi,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,e^c=ei,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6684391);