Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,11)(7,12,8,10)(15,21)(16,23)(18,25)(20,26)(22,27)(24,28);(2,3)(5,6)(7,10)(8,12)(9,11)(13,14)(15,22)(16,20)(17,19)(18,24)(21,27)(23,26)(25,28);(1,2,4,3)(13,15,23,28,17,24,16,21)(14,18,26,27,19,22,20,25);(5,7,9,8)(6,10,11,12)(13,14)(15,22)(16,20)(17,19)(18,24)(21,27)(23,26)(25,28);(1,3,4,2)(5,8,9,7)(6,10,11,12)(14,19)(18,22)(20,26)(25,27)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281016, #72284578, #72284598, … |
| maximal normal subgroup of | #72281016, #72284578, #72284598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724222);