Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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×#3513 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,26,32)(23,29,35,36)(24,31,25,30)(27,33,28,34);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,26,35)(22,27,32,28)(24,33,25,34)(29,30,36,31);(7,13)(10,17)(14,19)(18,20)(21,23)(22,28)(24,33)(25,34)(26,35)(27,32)(29,30)(31,36);(7,13)(10,17)(14,19)(18,20)(21,23)(22,29)(24,34)(25,33)(26,35)(27,31)(28,30)(32,36);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,24,26,25)(22,30,32,31)(23,33,35,34)(27,36,28,29)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72256862, #72256965, #72257082, … |
| maximal normal subgroup of | #72256862, #72256965, #72257082, … |
| maximal quotient of | #72308790 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6403481);