Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60753 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3184 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22)(23,30)(24,28)(25,33)(26,29)(27,31)(32,35)(34,36);(1,3,4,2)(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,22)(23,27)(24,33)(25,28)(26,29)(30,31)(32,36)(34,35);(7,14)(10,13)(15,20)(17,19)(21,23,24,34,26,32,25,31)(22,27,33,35,29,36,28,30);(6,11)(10,13)(12,18)(17,19)(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,35,36,30);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)(23,32)(27,36)(30,35)(31,34)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^b=di,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240753, #72250952, #72257061, … |
| maximal normal subgroup of | #72240753, #72250952, #72257061, … |
| maximal quotient of | #72240773 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283753);