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 | |
4083 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)(7,17)(8,13)(9,16)(10,14)(11,15)(12,19)(18,32)(20,27)(21,26)(22,29)(23,31)(24,30)(25,35)(28,33)(34,36);(5,7,11,21)(6,12,15,27)(8,18,23,34)(9,20,10,19)(13,28,31,25)(14,26,16,17)(22,35,24,33)(29,32,30,36);(2,4)(5,8)(6,13)(7,18)(9,24)(10,22)(11,23)(12,25)(14,29)(15,31)(16,30)(17,32)(19,35)(20,33)(21,34)(26,36)(27,28);(6,14)(9,10)(12,26)(13,29)(15,16)(17,27)(19,20)(22,24)(25,36)(28,32)(30,31)(33,35)Polycyclic presentation:
2^9|b^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 | #72238124, #72239541, #72251087, … |
| maximal normal subgroup of | #72238124, #72239541, #72251087, … |
| maximal quotient of | #72238125, #72239551, #72257318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301300);