Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2046 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)(7,16,19,11)(8,13)(9,12)(10,14)(15,23)(17,30,20,25)(18,26,32,24)(21,29)(22,27)(28,34)(31,36,33,35);(1,2,3,4)(5,7)(6,11)(8,20)(9,18)(10,19)(12,24)(13,25)(14,16)(15,26)(17,21)(22,33)(23,32)(27,35)(28,36)(29,30)(31,34);(6,12)(9,23)(11,24)(13,27)(14,15)(16,26)(18,32)(22,34)(25,35)(28,29)(30,36)(31,33);(5,8)(6,13)(7,17,19,20)(9,22)(10,21)(11,25,16,30)(12,27)(14,29)(15,28)(18,31,32,33)(23,34)(24,35,26,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=de,g^b=gi,f^c=fi,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72251062, #72251086, #72251087, … |
| maximal normal subgroup of | #72251062, #72251086, #72251087, … |
| maximal quotient of | #72313697, #72314558 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263699);