Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14187 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,8,13,6,12)(3,14,16,9,17,25,4,18)(5,20,21,10,22,27,23,26)(15,24,29,32,30,31,19,28);(1,3)(2,9)(4,6)(5,15)(7,16)(8,17)(10,24)(11,25)(12,14)(13,18)(19,23)(20,28)(21,29)(22,30)(26,31)(27,32);(1,4,6,17,8,16,7,3)(2,9,11,25,13,18,12,14)(5,19,23,30,22,29,21,15)(10,24,27,32,26,31,20,28);(1,5,7,21,8,22,6,23)(2,10,11,27,13,26,12,20)(3,15,16,29,17,30,4,19)(9,24,25,32,18,31,14,28)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215974, #72219684, #72219690, … |
| maximal normal subgroup of | #72215974, #72219684, #72219690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284142);