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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,26,31)(23,27,32,40)(24,29,35,43)(25,28,37,30)(33,39,45,50)(34,41,46,38)(36,42,48,44)(47,51,52,49);(2,3)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23)(22,27)(24,33)(25,34)(26,32)(28,41)(29,39)(30,38)(31,40)(35,45)(36,47)(37,46)(42,51)(43,50)(44,49)(48,52);(1,2,4,3)(22,28)(23,32)(25,37)(27,38)(29,42)(30,31)(33,45)(36,48)(39,49)(40,41)(43,44)(50,51);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,24)(22,29)(23,33)(25,36)(26,35)(27,39)(28,42)(30,44)(31,43)(32,45)(34,47)(37,48)(38,49)(40,50)(41,51)(46,52);(1,3,4,2)(7,15)(10,13)(14,20)(18,19)(23,32)(24,35)(27,40)(29,43)(34,46)(36,48)(38,41)(42,44)Polycyclic presentation:
2^9|a^2=fi,c^2=hi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6716128);