Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13291 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1993 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(6,12,14,15)(8,21)(9,23)(10,22)(13,28,30,25)(16,31,26,27)(17,32,33,36)(18,34)(19,35)(24,29);(5,6,9,15,10,16,23,31,11,14,8,12,22,26,21,27)(7,17,18,32,19,30,34,28,20,33,24,36,35,13,29,25);(2,4)(5,7)(6,13,14,30)(8,18)(9,24)(10,19)(11,20)(12,25,15,28)(16,17,26,33)(21,34)(22,35)(23,29)(27,32,31,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,c^b=ch,e^b=ei,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249949, #72249952, #72249962, … |
| maximal normal subgroup of | #72249949, #72249952, #72249962, … |
| maximal quotient of | #72265682, #72279631 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54390);