Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34977 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,20,16,19,22,18,17)(15,23,28,21,26,27,25,24);(1,2)(3,4)(13,15,19,26)(14,21,22,24)(16,23,17,27)(18,28,20,25);(2,4)(6,9)(10,12)(13,16,20,22,19,17,18,14)(15,21,28,27,26,24,25,23);(2,4)(13,17,18,22,19,16,20,14)(15,24,25,27,26,21,28,23);(5,7,8,11)(6,10,9,12)(14,22)(16,17)(21,24)(23,27)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fi,d^2=f,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72281632, #72282279, #72282435, … |
| maximal normal subgroup of | #72281632, #72282279, #72282435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363743);