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×#3025 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2204 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)(5,6,10,14)(7,15,18,28)(8,19,21,12)(9,22,24,13)(11,23,26,17)(16,30,32,29)(20,27,36,34)(25,31,33,35);(5,7)(6,11)(8,16)(9,23)(10,18)(12,25)(13,28)(14,26)(15,22)(17,24)(19,33)(20,35)(21,32)(27,30)(29,34)(31,36);(2,4)(9,24)(13,22)(20,36)(27,34);(5,8,10,21)(6,12,14,19)(7,16,18,32)(9,20,24,36)(11,25,26,33)(13,27,22,34)(15,29,28,30)(17,31,23,35)Polycyclic presentation:
2^9|a^2=i,d^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 | #72256635, #72256636, #72256645, … |
| maximal normal subgroup of | #72256635, #72256636, #72256645, … |
| maximal quotient of | #72263811, #72263816, #72263830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406126);