Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484139 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3050 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,11,16,29,23,12,17,26,9,8,14,24,27)(7,18,21,34,19,25,28,13,22,31,35,20,30,33,36,32);(5,7,12,22)(6,13)(8,19,11,30)(9,25)(10,28,26,36)(14,20)(15,33)(16,34)(17,32)(18,27)(21,24,35,29)(23,31);(2,4)(5,8,12,11)(6,14,17,16)(7,19,22,30)(9,23,15,27)(10,24,26,29)(13,20,32,34)(18,25,31,33)(21,28,35,36);(5,9,26,14,11,27,24,6,12,15,10,16,8,23,29,17)(7,20,35,33,19,32,36,18,22,34,21,25,30,13,28,31)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=i,d^2=ei,e^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,d^b=de,e^b=eh,g^b=gh,h^b=hi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286580, #72286631, #72286745, … |
| maximal normal subgroup of | #72286580, #72286631, #72286745, … |
| maximal quotient of | #72287105, #72287209, #72287414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405574);