Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
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×#3005 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1988 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)(8,13)(9,21)(10,24)(11,15)(12,17)(14,20)(18,35)(19,29)(22,31)(23,30)(25,28)(26,32)(27,36)(33,34);(5,7)(6,12)(8,17)(9,22)(10,25)(11,19)(13,29)(14,31)(15,28)(16,24)(18,20)(21,27)(23,34)(26,36)(30,35)(32,33);(5,8,11,10)(6,13,15,24)(7,17,19,25)(9,20,23,26)(12,16,28,29)(14,30,32,21)(18,34,36,22)(27,33,35,31);(5,9,11,23)(6,14,15,32)(7,18,19,36)(8,20,10,26)(12,27,28,35)(13,30,24,21)(16,33,29,31)(17,34,25,22)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72238736, #72264989, #72265017, … |
| maximal normal subgroup of | #72238736, #72264989, #72265017, … |
| maximal quotient of | #72238748 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290344);