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 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2658 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)(7,16)(8,12)(9,14)(10,13)(11,18)(15,23)(17,30)(19,26)(20,24)(21,27)(22,28)(25,32)(29,34)(31,36)(33,35);(5,7)(6,11)(8,20,22,17)(9,18)(10,19)(12,24,28,30)(13,25)(14,16)(15,26)(21,33,34,31)(23,32)(27,35,29,36);(2,4)(5,8)(6,12)(7,17)(9,22)(10,21)(11,24)(13,27)(14,28)(15,29)(16,30)(18,20)(19,31)(23,34)(25,35)(26,36)(32,33);(6,13)(10,23)(11,25)(12,27)(14,15)(16,26)(19,32)(21,34)(24,35)(28,29)(30,36)(31,33)Polycyclic presentation:
2^9|b^2=ei,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 | #72237694, #72251058, #72251087, … |
| maximal normal subgroup of | #72237694, #72251058, #72251087, … |
| maximal quotient of | #72314520, #72314522 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301556);