Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234345 |
| 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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1375 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)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,29,34)(27,36,38,31)(28,33,41,52)(30,44,43,35)(32,45,50,42)(37,51,40,47)(39,48,55,53)(46,54,49,56);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,31)(28,40)(29,38)(30,39)(32,49)(33,51)(34,36)(35,48)(37,41)(42,56)(43,55)(44,53)(45,54)(46,50)(47,52);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28)(26,32)(27,37)(29,41)(30,45)(31,46)(33,44)(34,50)(35,52)(36,49)(38,40)(39,56)(42,43)(47,53)(48,51)(54,55);(3,8)(5,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,37)(29,41)(30,42)(31,47)(32,44)(34,52)(35,50)(36,51)(38,40)(39,54)(43,45)(46,53)(48,49)(55,56);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(26,34)(27,38)(28,41)(35,44)(39,55)(42,45)(46,49)(47,51)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7145327);