Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300980 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1639 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(17,18,23,27)(19,28,32,42)(20,25,33,39)(21,26,22,36)(24,30,38,31)(29,43,44,48)(34,40,35,41)(37,46,47,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(20,33)(25,39)(29,44)(34,35)(37,47)(40,41)(43,48)(45,46);(9,11)(10,13)(12,16)(14,15)(17,19,23,32)(18,24,27,38)(20,29,33,44)(21,31,22,30)(25,37,39,47)(26,42,36,28)(34,46,35,45)(40,43,41,48);(3,8)(5,7)(10,14)(11,16)(17,20)(18,25)(19,29)(21,35)(22,34)(23,33)(24,37)(26,41)(27,39)(28,43)(30,46)(31,45)(32,44)(36,40)(38,47)(42,48);(2,6)(5,7)(10,14)(13,15)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,34,33,35)(24,28,38,42)(25,40,39,41)(29,45,44,46)(37,48,47,43)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7208453);