Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394102 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4039 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)(7,13)(8,20)(9,12)(10,15)(11,16)(14,22)(17,31)(18,27)(19,29)(21,30)(23,26)(24,25)(28,32)(33,35)(34,36);(6,12)(9,10)(13,27)(14,26)(15,16)(18,24)(20,30)(21,22)(25,29)(28,31)(32,34)(35,36);(2,4)(5,7)(6,13)(8,17)(9,24)(10,18)(11,19)(12,25)(14,28)(15,27)(16,29)(20,31)(21,34)(22,32)(23,33)(26,35)(30,36);(5,8,11,23)(6,14,16,30)(7,17,19,33)(9,22,10,21)(12,26,15,20)(13,28,29,36)(18,34,24,32)(25,35,27,31)Polycyclic presentation:
2^9|d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72251089, #72251128, #72255889, … |
| maximal normal subgroup of | #72251089, #72251128, #72255889, … |
| maximal quotient of | #72257298, #72257301, #72257744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301299);