Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499787 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1753 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,16)(11,17,19,24)(12,15,20,23)(14,21,22,18);(1,3,4,8)(2,5,6,7)(12,20)(15,23);(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,21)(23,24);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=i,b^2=e,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fh,h^c=hi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72238456, #72240104, #72249585, … |
| maximal normal subgroup of | #72238456, #72240104, #72249585, … |
| maximal quotient of | #72238460, #72240141, #72262502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407249);