Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,25)(21,26)(22,35)(23,27)(24,30)(29,43)(31,38)(32,40)(33,41)(34,42)(36,39)(37,46)(44,48)(45,47);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(20,32)(25,40)(29,44)(33,34)(37,47)(41,42)(43,48)(45,46);(11,15)(13,16)(17,19,23,31)(18,24,27,39)(20,29,32,44)(21,30,22,36)(25,37,40,47)(26,38,35,28)(33,45,34,46)(41,43,42,48);(3,8)(5,7)(10,14)(11,15)(17,20)(18,25)(19,29)(21,34)(22,33)(23,32)(24,37)(26,42)(27,40)(28,43)(30,46)(31,44)(35,41)(36,45)(38,48)(39,47);(2,6)(5,7)(11,15)(13,16)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,33,32,34)(24,38,39,28)(25,41,40,42)(29,45,44,46)(37,43,47,48)Polycyclic presentation:
2^9|c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7107305);