Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#390 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,17)(8,22,26,13)(9,16)(10,14)(11,15)(12,20)(18,33,34,30)(19,29)(21,28)(23,36,35,27)(24,31,25,32);(1,2,3,4)(5,7,9,20,11,19,10,21)(6,12,14,17,15,28,16,29)(8,23,24,18,26,35,25,34)(13,30,31,27,22,33,32,36);(5,8)(6,13)(7,18)(9,25)(10,24)(11,26)(12,27)(14,32)(15,22)(16,31)(17,30)(19,34)(20,23)(21,35)(28,36)(29,33);(7,19)(12,28)(17,29)(18,34)(20,21)(23,35)(27,36)(30,33)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237649, #72237652, #72237667, … |
| maximal normal subgroup of | #72237649, #72237652, #72237667, … |
| maximal quotient of | C2×#357809, #72314670 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265005);