Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518952 |
| 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×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2229 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,10)(7,11,8,12)(13,14,17,20)(15,25,26,22)(16,21,27,28)(18,24,19,23);(1,2)(3,4)(6,10)(11,12)(14,20)(15,16)(18,19)(21,22)(25,28)(26,27);(5,7,9,8)(6,11,10,12)(13,15,18,16,17,26,19,27)(14,21,23,25,20,28,24,22);(2,4)(13,16,18,26,17,27,19,15)(14,22,23,21,20,25,24,28);(5,8,9,7)(6,11,10,12)(14,20)(15,26)(16,27)(23,24)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72263848, #72263850, #72263878, … |
| maximal normal subgroup of | #72263848, #72263850, #72263878, … |
| maximal quotient of | #72265089 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472123);