Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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×#3565 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2754 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)(21,22)(23,27)(24,29)(25,28)(26,31)(30,37)(32,40)(33,39)(34,41)(35,44)(36,42)(38,46)(43,48)(45,51)(47,50)(49,52);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,26,32)(22,27,31,40)(24,33,35,45)(25,34,37,46)(28,41,30,38)(29,39,44,51)(36,47,48,52)(42,50,43,49);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(22,28)(23,32)(24,35)(25,37)(27,38)(29,43)(30,31)(39,50)(40,41)(42,44)(47,52)(49,51);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,24)(22,29)(23,33)(25,36)(26,35)(27,39)(28,42)(30,43)(31,44)(32,45)(34,47)(37,48)(38,49)(40,51)(41,50)(46,52);(6,12)(10,13)(11,20)(18,19)(23,32)(24,35)(27,40)(29,44)(34,46)(36,48)(38,41)(42,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436844);