Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1776 |
| prime radical | C24 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
520 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)(5,6)(7,20,23,36)(8,25,24,33)(9,27)(10,30)(11,18)(12,32)(13,19)(14,21,15,26)(16,28)(17,29)(22,34,31,35);(2,4)(5,7,11,24,13,23,12,8)(6,14,18,34,19,15,32,35)(9,22,10,21,29,31,28,26)(16,33,27,36,30,25,17,20);(5,8,12,23,13,24,11,7)(6,15,18,35,19,14,32,34)(9,21,28,22,29,26,10,31)(16,33,27,36,30,25,17,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,d^b=dhi,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72320970, #72321008, #72321042, … |
| maximal normal subgroup of | #72320970, #72321008, #72321042, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54498);