Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499792 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3547 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,17)(14,16)(15,18)(19,20)(21,22)(23,32)(24,37)(25,30)(26,42)(27,31)(28,35)(29,39)(33,50)(34,44)(36,48)(38,49)(40,47)(41,46)(43,52)(45,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,27,36)(22,28,31,44)(24,38,39,33)(25,35,26,34)(29,45,37,43)(30,48,42,32)(40,51,41,52)(46,50,47,49);(2,4)(6,11)(7,13)(8,15)(16,20)(21,24)(22,29)(23,33)(25,41)(26,40)(27,39)(28,43)(30,47)(31,37)(32,49)(34,52)(35,51)(36,38)(42,46)(44,45)(48,50);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,16,17,20)(24,39)(29,37)(33,38)(40,41)(43,45)(46,47)(49,50)(51,52);(7,13)(10,17)(14,19)(16,20)(21,25,27,26)(22,30,31,42)(23,34,36,35)(24,40,39,41)(28,32,44,48)(29,46,37,47)(33,51,38,52)(43,50,45,49)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6432048);