Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222240 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2395 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)(8,20,24,13)(9,25)(10,27)(11,16)(12,19)(14,26)(15,28)(18,31,21,34)(22,32,29,35)(23,30,36,33);(5,7,11,14)(6,12,16,9)(8,21)(10,25,28,19)(13,29)(15,26,27,17)(18,24)(20,22)(23,32)(30,34)(31,33)(35,36);(2,4)(5,8)(6,13)(7,18)(9,22)(10,23)(11,24)(12,29)(14,21)(15,30)(16,20)(17,31)(19,32)(25,35)(26,34)(27,33)(28,36);(5,9,11,12)(6,14,16,7)(8,22,24,29)(10,17,28,26)(13,21,20,18)(15,19,27,25)(23,31,36,34)(30,32,33,35)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72314237, #72314265, #72314335, … |
| maximal normal subgroup of | #72314237, #72314265, #72314335, … |
| maximal quotient of | C2×#364632, #72314517 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271828);