Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #23810 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(8,16)(11,17)(15,20)(18,19)(21,22,26,30)(23,24,33,28)(25,29,35,34)(27,36,32,31);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23)(22,28)(24,30)(25,27)(26,33)(29,31)(32,35)(34,36);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,27,34,26,28,32,29)(22,25,31,33,30,35,36,23);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,25)(22,29)(23,32)(24,31)(26,35)(27,33)(28,36)(30,34);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(22,30)(23,33)(25,35)(31,36)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=g,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72314172 |
| maximal normal subgroup of | #72314172 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093763);