Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35537 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(21,22)(23,27)(24,26)(25,29)(28,32)(30,34)(31,33)(35,36);(1,2,3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(22,26)(24,32)(27,33)(28,29)(31,36)(34,35);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,23,25,30)(22,27,29,34)(24,31,32,36)(26,33,28,35);(6,12)(10,13)(11,20)(18,19)(23,30)(27,34)(31,36)(33,35)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257144, #72257176, #72257220, … |
| maximal normal subgroup of | #72257144, #72257176, #72257220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6437128);