Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14015 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,25)(3,9,16,5,11,23,15,24)(4,17,22,29,20,31,21,10)(14,19,18,32,28,27,30,26);(1,3,7,16,8,11,6,15)(2,9,12,5,13,23,25,24)(4,18,22,28,20,30,21,14)(10,27,17,26,29,19,31,32);(1,4,6,21,8,20,7,22)(2,10,12,17,13,29,25,31)(3,14,16,18,11,28,15,30)(5,19,9,26,24,27,23,32);(1,5)(2,11)(3,13)(4,19)(6,9)(7,23)(8,24)(10,28)(12,15)(14,29)(16,25)(17,30)(18,31)(20,27)(21,26)(22,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285688, #72285720, #72286225, … |
| maximal normal subgroup of | #72285688, #72285720, #72286225, … |
| maximal quotient of | C2×#368091, #72316853, #72316946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275287);