Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324564 |
| 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 | #53152 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,4)(3,7)(9,10,14,17)(11,18,20,15)(12,16,13,22)(19,24,21,23)(25,26)(27,35)(28,39)(29,33)(30,34)(31,38)(32,36)(37,48)(40,55)(41,51)(42,45)(43,56)(44,50)(46,52)(47,53)(49,54);(1,2)(3,8)(4,5)(6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24)(25,27,30,38)(26,31,34,35)(28,36,41,53)(29,42,44,37)(32,39,47,51)(33,48,50,45)(40,54,43,52)(46,56,49,55);(1,3)(2,6)(4,8)(5,7)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28,38,53,30,41,27,36)(26,32,35,51,34,47,31,39)(29,43,42,52,44,40,37,54)(33,49,48,55,50,46,45,56);(3,7)(6,8)(11,20)(15,18)(19,21)(23,24)(25,29)(26,33)(27,37)(28,40)(30,44)(31,45)(32,46)(34,50)(35,48)(36,52)(38,42)(39,55)(41,43)(47,49)(51,56)(53,54);(1,4,5,2)(3,8,7,6)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(26,34)(29,44)(31,35)(32,47)(37,42)(39,51)(40,43)(52,54)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=fg,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7262613);