Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2428 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14012 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,16,25)(4,18,20,27)(5,11,17,24)(6,13)(7,14)(8,12)(10,21,28,22)(15,26)(19,31)(23,30)(29,32);(1,3,7,17,8,16,6,5)(2,9,13,11,12,25,14,24)(4,19,22,29,20,30,21,15)(10,26,18,23,28,32,27,31);(1,4)(2,10)(3,15)(5,19)(6,22)(7,21)(8,20)(9,23)(11,26)(12,28)(13,27)(14,18)(16,29)(17,30)(24,32)(25,31);(1,5,6,16,8,17,7,3)(2,11,14,9,12,24,13,25)(4,19,22,29,20,30,21,15)(10,26,18,23,28,32,27,31)Polycyclic presentation:
2^9|a^2=h,b^2=egh,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319753 |
| maximal normal subgroup of | #72319753 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275136);