Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996607 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,10,11)(7,9,12,8)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26);(2,3)(15,21)(16,24)(18,25)(19,23);(5,7,9,11,10,12,8,6)(13,15,17,21)(14,18,20,25)(16,22,24,27)(19,26,23,28);(1,3,4,2)(15,21)(18,25)(22,27)(26,28);(5,8,10,9)(6,7,11,12)(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)Polycyclic presentation:
2^9|a^2=fhi,c^2=gh,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283894, #72284171, #72284466, … |
| maximal normal subgroup of | #72283894, #72284171, #72284466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104049);