Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470137 |
| 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 | #37550 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2915 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)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22)(23,27)(24,34)(25,37)(26,29)(28,35)(30,39)(31,45)(32,41)(33,49)(36,44)(38,43)(40,47)(42,48)(46,52)(50,51);(2,4)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23)(22,27)(24,31)(25,33)(26,32)(28,40)(29,41)(30,42)(34,45)(35,47)(36,46)(37,49)(38,50)(39,48)(43,51)(44,52);(7,13)(10,17)(14,15)(18,19)(21,24,26,36)(22,28,29,43)(23,31,32,46)(25,38,39,35)(27,40,41,51)(30,44,37,34)(33,50,48,47)(42,52,49,45);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(22,29)(27,41)(28,43)(30,37)(34,44)(40,51)(42,49)(45,52);(6,12)(10,17)(11,20)(18,19)(23,32)(27,41)(31,46)(33,48)(40,51)(42,49)(45,52)(47,50)Polycyclic presentation:
2^9|c^2=i,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388607);