Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#3551 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2417 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)(5,6,11,17,12,18,9,15)(7,19,22,35,23,34,21,29)(8,16,24,14,10,26,25,33)(13,28,30,27,32,36,31,20);(5,7)(6,13)(8,20)(9,22)(10,27)(11,21)(12,23)(14,29)(15,31)(16,34)(17,30)(18,32)(19,26)(24,36)(25,28)(33,35);(2,4)(5,8,12,10)(6,14,18,33)(7,20,23,27)(9,24,11,25)(13,29,32,35)(15,26,17,16)(19,30,34,31)(21,28,22,36);(5,9,12,11)(6,15,18,17)(7,21,23,22)(8,24,10,25)(13,30,32,31)(14,26,33,16)(19,29,34,35)(20,28,27,36)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261190, #72261267, #72261282, … |
| maximal normal subgroup of | #72261190, #72261267, #72261282, … |
| maximal quotient of | #72250630 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400812);