Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #368859 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14023 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,13,19,11)(3,14,17,28,18,27,16,8)(4,15,22,9,21,29,20,30)(5,23,25,31,26,32,24,10);(1,3,6,17,7,18,19,16)(2,8,12,14,13,28,11,27)(4,10,22,23,21,31,20,32)(5,15,25,9,26,29,24,30);(1,4,19,22,7,21,6,20)(2,9,12,15,13,30,11,29)(3,10,16,23,18,31,17,32)(5,14,24,28,26,27,25,8);(1,5,7,26)(2,10,13,31)(3,15,18,29)(4,14,21,27)(6,25,19,24)(8,22,28,20)(9,16,30,17)(11,32,12,23)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72215931, #72216144, #72219683, … |
| maximal normal subgroup of | #72215931, #72216144, #72219683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276058);