Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3041 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,16)(8,12)(9,13)(10,15)(11,18)(14,24)(17,32)(19,30)(20,31)(21,27)(22,29)(23,26)(25,34)(28,35)(33,36);(1,3,4,2)(5,7)(6,11)(8,20)(9,23)(10,19)(12,25)(13,16)(14,30)(15,26)(17,22)(18,24)(21,33)(27,31)(28,32)(29,36)(34,35);(5,8)(6,12,15,29)(7,17,19,20)(9,21)(10,22)(11,25)(13,27,14,28)(16,31)(18,33,23,34)(24,35)(26,36)(30,32);(6,13)(9,24)(11,16)(12,27)(14,15)(18,23)(21,35)(25,31)(26,30)(28,29)(32,36)(33,34)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fgi,e^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,f^c=fi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72255945, #72262411, #72262421, … |
| maximal normal subgroup of | #72255945, #72262411, #72262421, … |
| maximal quotient of | #72313444, #72314552 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276061);