Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,37)(28,34)(29,36)(30,32)(31,35)(33,39)(38,53)(40,52)(41,47)(42,56)(43,51)(44,50)(45,46)(48,55)(49,54);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,32)(28,41)(33,46)(35,36)(37,52)(38,42)(43,44)(54,55);(1,2,5,4)(3,6,8,7)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,27)(26,33)(28,42)(29,39)(30,40)(31,45)(32,46)(34,49)(35,37)(36,52)(38,41)(43,55)(44,54)(47,48)(50,53)(51,56);(1,4,5,2)(3,6,8,7)(12,20)(15,24)(17,19)(21,23)(25,28)(26,34)(27,38)(29,44)(30,41)(31,43)(32,47)(33,48)(35,51)(36,50)(37,53)(39,55)(40,42)(45,54)(46,49)(52,56);(10,16)(11,18)(15,24)(17,19)(25,29,30,31)(26,35,32,36)(27,39,40,45)(28,43,41,44)(33,37,46,52)(34,50,47,51)(38,54,42,55)(48,56,49,53)Polycyclic presentation:
2^9|a^2=fhi,c^2=fhi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 7012790);