Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419895 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1605 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15)(7,16,20,12)(8,21)(9,11,24,14)(13,28)(17,25,19,27)(18,33)(22,30)(23,29)(26,32)(31,34)(35,36);(1,2)(3,4)(6,11)(7,17)(8,22)(9,24)(14,15)(19,20)(21,29)(23,28)(25,27)(26,36)(31,33)(32,34);(2,4)(5,7,10,20)(6,12,15,16)(8,18,23,35)(9,19,24,17)(11,25,14,27)(13,26,30,31)(21,33,29,36)(22,34,28,32);(5,8,9,22,10,23,24,28)(6,13,14,29,15,30,11,21)(7,18,19,34,20,35,17,32)(12,26,27,36,16,31,25,33)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285083, #72285470, #72285552, … |
| maximal normal subgroup of | #72285083, #72285470, #72285552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327092);