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 | #3147 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,11,17,12,18,9,15)(7,13,10,26,21,16,20,30)(8,22,24,35,25,31,28,29)(14,27,32,19,34,23,33,36);(2,3)(5,7,11,10,12,21,9,20)(6,13,17,26,18,16,15,30)(8,23,24,36,25,27,28,19)(14,31,32,29,34,22,33,35);(5,8,12,25)(6,14,18,34)(7,19,21,36)(9,24,11,28)(10,27,20,23)(13,29,16,35)(15,32,17,33)(22,26,31,30);(1,3,4,2)(5,9,12,11)(6,15,18,17)(7,20,21,10)(8,24,25,28)(13,30,16,26)(14,32,34,33)(19,23,36,27)(22,35,31,29)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,d^2=ei,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ch,d^b=de,f^b=fi,f^c=fhi,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72286631, #72286658, #72320347, … |
| maximal normal subgroup of | #72286631, #72286658, #72320347, … |
| maximal quotient of | #72287604, #72287756, #72309469, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405533);