Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30441 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| 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 | |
1435 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)(7,16)(8,21)(9,11)(10,15)(12,19)(13,23)(14,24)(17,29)(18,31)(20,26)(22,27)(25,30)(28,33)(32,35)(34,36);(6,11)(7,17)(8,22)(9,24)(13,21)(14,15)(16,26)(19,20)(23,25)(28,31)(32,33)(35,36);(5,7,10,20)(6,12)(8,18,23,34)(9,19,24,17)(11,26)(13,28)(14,16)(15,29)(21,35)(22,32,25,33)(27,36)(30,31);(5,8,10,23)(6,13,15,21)(7,18,20,34)(9,25,24,22)(11,27,14,30)(12,28,29,35)(16,31,26,36)(17,32,19,33)Polycyclic presentation:
2^9|a^2=h,c^2=eg,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250584, #72256515, #72256521 |
| maximal normal subgroup of | #72250584, #72256515, #72256521 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293124);