Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23895 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14171 |
| prime radical | #23443 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,16,22)(3,7,13,19,12,17,26,28)(4,8,15,21,14,20,27,31)(11,23,25,32,24,18,29,30);(7,17)(18,23)(19,28)(30,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,18)(9,17)(10,19)(14,24)(15,25)(16,26)(20,23)(21,30)(22,28)(27,29)(31,32);(1,4,5,15,6,14,16,27)(2,8,10,21,9,20,22,31)(3,11,13,25,12,24,26,29)(7,18,19,30,17,23,28,32)Polycyclic presentation:
2^9|a^2=h,d^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,e^c=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#375777, #72237690, #72240435, … |
| maximal normal subgroup of | C2×#375777, #72237690, #72240435, … |
| maximal quotient of | C2×#375777, #72251323, #72262339, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282973);