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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,30)(24,28)(25,35)(26,29)(27,33)(31,32)(34,36);(1,2)(3,4)(7,13)(8,16)(10,17)(11,15);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,25,33,26,34,24,32)(22,27,28,30,29,31,35,36);(2,4)(7,13)(10,17)(14,19)(18,20)(21,22)(23,31)(24,35)(25,28)(26,29)(27,34)(30,32)(33,36);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,24,26,25)(22,28,29,35)(23,32,34,33)(27,30,31,36)Polycyclic presentation:
2^9|a^2=h,c^2=g,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 | #72257063, #72257108, #72257220, … |
| maximal normal subgroup of | #72257063, #72257108, #72257220, … |
| maximal quotient of | #72265484 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7080551);