Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484039 |
| 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 | |
739 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)(6,11,13,27)(8,19)(9,23)(12,26,28,33)(14,30,24,20)(15,29,32,31)(16,34)(17,25)(21,35)(22,36);(5,6,9,24,10,13,23,14)(7,15,17,26,18,32,25,33)(8,20,19,11,22,30,36,27)(12,21,31,16,28,34,29,35);(2,4);(5,7,10,18)(6,12,13,28)(8,21,22,34)(9,25,23,17)(11,26,27,33)(14,31,24,29)(15,20,32,30)(16,19,35,36)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=i,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=de,e^b=eh,g^b=gi,h^b=hi,e^d=ehi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286631, #72286724, D4×#2180, … |
| maximal normal subgroup of | #72286631, #72286724, D4×#2180, … |
| maximal quotient of | #72287643, #72287772, #72309464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405313);