Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3858 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)(7,17,20,13,21,28,19,26)(8,22,23,30,24,14,25,29)(18,32,33,27,34,31,35,36);(2,4)(6,12)(9,10)(14,22)(17,28)(19,20)(23,25)(31,32)(33,35);(5,7,11,21)(6,13,12,26)(8,18,24,34)(9,20,10,19)(14,27,22,36)(15,28,16,17)(23,35,25,33)(29,32,30,31);(2,4)(5,8)(6,14)(7,18)(9,23)(10,25)(11,24)(12,22)(13,27)(15,29)(16,30)(17,31)(19,33)(20,35)(21,34)(26,36)(28,32)Polycyclic presentation:
2^9|a^2=h,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,f^c=fi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72261616, #72261630, #72261697, … |
| maximal normal subgroup of | #72261616, #72261630, #72261697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266802);