Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1331 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,16,11,12,9,15)(7,17,20,26,18,30,25,13)(8,21,22,14,23,29,24,27)(19,28,34,31,32,35,33,36);(6,12)(7,18)(9,10)(17,30)(21,29)(22,24)(31,36)(33,34);(2,4)(5,7,10,25,11,18,9,20)(6,13,15,17,12,26,16,30)(8,19,22,33,23,32,24,34)(14,28,21,31,27,35,29,36);(2,4)(5,8,11,23)(6,14,12,27)(7,19,18,32)(9,24,10,22)(13,28,26,35)(15,21,16,29)(17,31,30,36)(20,34,25,33)Polycyclic presentation:
2^9|a^2=h,c^2=eh,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72287280, #72287286, #72287360, … |
| maximal normal subgroup of | #72287280, #72287286, #72287360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295563);