Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23892 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3781 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)(5,6,10,15)(7,16,20,32)(8,12,23,31)(9,24,26,14)(11,25,28,19)(13,22,30,21)(17,29,35,33)(18,27,34,36);(5,7)(6,11)(8,18)(9,25)(10,20)(12,29)(13,27)(14,32)(15,28)(16,24)(17,22)(19,26)(21,35)(23,34)(30,36)(31,33);(5,8)(6,12)(7,17)(9,21)(10,23)(11,27)(13,24)(14,30)(15,31)(16,33)(18,25)(19,34)(20,35)(22,26)(28,36)(29,32);(5,8)(6,13)(7,18)(9,22)(10,23)(11,27)(12,24)(14,31)(15,30)(16,29)(17,25)(19,35)(20,34)(21,26)(28,36)(32,33)Polycyclic presentation:
2^9|a^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72312189, #72312195, #72312227, … |
| maximal normal subgroup of | #72312189, #72312195, #72312227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266879);