Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456658 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1185 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)(7,16,18,11)(8,12)(9,15)(10,13)(14,24)(17,26,34,33)(19,27,20,28)(21,36,35,29)(22,31)(23,32)(25,30);(5,7)(6,11,15,16)(8,21)(9,18)(10,20)(12,29,31,36)(13,27,14,28)(17,23)(19,24)(22,35)(25,34)(26,32,33,30);(2,4)(5,8,9,22)(6,12,15,31)(7,17,18,34)(10,25,24,23)(11,26,16,33)(13,30,14,32)(19,35,20,21)(27,36,28,29);(6,13)(8,22)(10,24)(11,27)(12,30)(14,15)(16,28)(19,20)(21,35)(26,29)(31,32)(33,36)Polycyclic presentation:
2^9|a^2=f,b^2=egh,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285161, #72285470, #72285522, … |
| maximal normal subgroup of | #72285161, #72285470, #72285522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379088);