Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1173 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,14,10,15,9,16)(7,17,19,12,20,29,21,28)(8,22,26,32,25,13,24,31)(18,30,23,36,34,33,35,27);(2,4)(5,7,10,20)(6,12,15,28)(8,23)(9,21,11,19)(13,30)(14,29,16,17)(18,24)(22,33)(25,35)(26,34)(27,32)(31,36);(1,2,3,4)(5,8,10,25)(6,13,15,22)(7,18,20,34)(9,24,11,26)(12,27,28,36)(14,31,16,32)(17,33,29,30)(19,23,21,35);(5,7)(6,12)(8,18)(9,21)(10,20)(11,19)(13,27)(14,29)(15,28)(16,17)(22,36)(23,26)(24,35)(25,34)(30,31)(32,33)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280214, #72280601, #72280718, … |
| maximal normal subgroup of | #72280214, #72280601, #72280718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274699);