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×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,8,4,6)(10,13)(11,15)(17,18,21,23,22,24,20,19);(10,13)(11,15)(17,19)(18,20)(21,24)(22,23);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16);(10,13)(14,16)(17,18,20,19,22,24,21,23);(1,4,5,3)(2,6,7,8)(9,11,12,15)(10,14,13,16)(17,20,22,21)(18,23,24,19)Polycyclic presentation:
2^9|a^2=fgh,c^2=gh,d^2=fi,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72251178, #72251252, #72251275, … |
| maximal normal subgroup of | #72251178, #72251252, #72251275, … |
| maximal quotient of | #72264723 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7025058);