Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| 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 | |
1321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17)(8,22,24,28)(9,13,26,23)(10,15)(11,16)(12,20)(14,29)(18,25,34,36)(19,33,35,32)(21,30)(27,31);(1,2)(3,4)(5,7,11,21)(6,12,16,31)(8,18,24,34)(9,25,26,36)(10,27,29,20)(13,32,23,33)(14,30,15,17)(19,28,35,22);(2,4)(5,8,10,13,11,24,29,23)(6,9,14,28,16,26,15,22)(7,18,20,33,21,34,27,32)(12,25,17,19,31,36,30,35);(5,9,11,26)(6,13,16,23)(7,19,21,35)(8,14,24,15)(10,28,29,22)(12,18,31,34)(17,33,30,32)(20,36,27,25)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=eh,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286523, #72286940, #72287067 |
| maximal normal subgroup of | #72286523, #72286940, #72287067 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295582);