Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #29607 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,25,30)(23,32,33,27)(24,36,38,29)(26,28,40,31)(34,42,37,45)(35,41,47,43)(39,46,50,44)(48,52,49,51);(2,3)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,27)(24,37)(25,33)(26,35)(28,43)(29,45)(30,32)(31,41)(34,38)(36,42)(39,49)(40,47)(44,51)(46,52)(48,50);(1,2,4,3)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(22,28)(23,33)(26,40)(27,41)(29,44)(30,31)(32,43)(34,37)(36,46)(39,50)(42,51)(45,52);(6,12)(10,13)(11,18)(17,19)(21,24,25,38)(22,29,30,36)(23,34,33,37)(26,39,40,50)(27,42,32,45)(28,44,31,46)(35,48,47,49)(41,51,43,52);(1,3,4,2)(7,15)(10,13)(14,20)(17,19)(24,38)(29,36)(34,37)(39,50)(42,45)(44,46)(48,49)(51,52)Polycyclic presentation:
2^9|a^2=i,c^2=fh,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6714942);