Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2291 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)(7,15)(8,12)(9,21)(10,13)(11,17)(14,24)(16,31)(18,29)(19,30)(20,27)(22,26)(23,28)(25,32)(33,36)(34,35);(5,7)(6,11)(8,16)(9,22)(10,18)(12,25)(13,26)(14,29)(15,21)(17,24)(19,34)(20,33)(23,32)(27,35)(28,36)(30,31);(2,4)(5,8,10,20)(6,12,13,27)(7,16,18,33)(9,23,24,19)(11,25,26,35)(14,30,21,28)(15,31,29,36)(17,32,22,34);(6,13)(11,26)(12,27)(14,21)(15,29)(25,35)(28,30)(31,36)Polycyclic presentation:
2^9|c^2=i,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,h^b=hi,e^c=eh,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72250024, #72250106, #72256600, … |
| maximal normal subgroup of | #72250024, #72250106, #72256600, … |
| maximal quotient of | #72250026, #72250099, #72256712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406124);