Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394358 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,19)(9,22)(10,13)(12,20)(14,24)(15,17)(16,31)(18,30)(21,28)(23,27)(25,33)(26,34)(29,35)(32,36);(5,6)(7,15,17,11)(8,12)(9,14)(10,13)(16,29,32,25)(18,30,34,26)(19,20)(21,27)(22,24)(23,28)(31,33,36,35);(2,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,25)(13,15)(14,26)(19,31)(20,33)(21,32)(22,30)(23,35)(24,34)(27,29)(28,36);(5,8,10,21)(6,12,13,27)(7,16,17,32)(9,23,24,20)(11,25,15,29)(14,28,22,19)(18,35,34,33)(26,36,30,31)Polycyclic presentation:
2^9|b^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72251058, #72251071, #72251089, … |
| maximal normal subgroup of | #72251058, #72251071, #72251089, … |
| maximal quotient of | #72314504, C2×#394359, #72314514 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301555);