Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324500 |
| 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 | #53073 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(9,10,14,16)(11,12,15,13)(17,18,23,27)(19,28,29,39)(20,25,32,40)(21,26,22,36)(24,34,37,31)(30,43,44,48)(33,41,35,42)(38,45,47,46);(1,2)(3,7)(4,6)(5,8)(9,11,14,15)(10,13,16,12)(19,29)(20,32)(24,37)(25,40)(28,39)(31,34)(33,35)(41,42);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,19,21,34,23,29,22,31)(18,24,36,39,27,37,26,28)(20,30,35,45,32,44,33,46)(25,38,41,48,40,47,42,43);(2,6)(3,8)(9,12,14,13)(10,15,16,11)(17,20)(18,25)(19,30)(21,35)(22,33)(23,32)(24,38)(26,42)(27,40)(28,43)(29,44)(31,46)(34,45)(36,41)(37,47)(39,48);(2,6)(5,7)(9,13,14,12)(10,15,16,11)(17,21,23,22)(18,26,27,36)(19,31,29,34)(20,33,32,35)(24,39,37,28)(25,41,40,42)(30,45,44,46)(38,43,47,48)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fgi,d^2=f,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=ehi,d^c=dhi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7246981);