Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #464072 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16500 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1479 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,8)(5,17)(6,10)(7,13)(9,18)(12,27)(14,26)(15,24)(16,23)(19,22)(20,30)(21,28)(25,29)(31,32);(1,3)(2,7,10,20)(4,12)(5,13)(6,14)(8,21,23,32)(9,11,17,26)(15,28)(16,29)(18,30)(19,31)(22,27,24,25);(4,15)(5,18)(7,20)(8,22)(9,17)(12,28)(13,30)(16,19)(21,25)(23,24)(27,32)(29,31);(1,4)(2,8)(3,12)(5,19)(6,16)(7,21)(9,24)(10,23)(11,25)(13,31)(14,29)(15,18)(17,22)(20,32)(26,27)(28,30)Polycyclic presentation:
2^9|b^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72215912, #72216013, #72216037, … |
| maximal normal subgroup of | #72215912, #72216013, #72216037, … |
| maximal quotient of | C2×#464072, #72255609, #72265539 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371268);