Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622467 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C23×#99 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C2×D4 | C25 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
4343 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,11,13,24)(4,8)(5,17,19,9)(6,10)(7,14)(12,25,26,32)(15,22)(16,29,28,23)(18,21)(20,27)(30,31);(1,3)(2,7)(4,12)(5,18)(6,13)(8,20)(9,24)(10,21)(11,17)(14,19)(15,26)(16,30)(22,31)(23,32)(25,29)(27,28);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,23)(10,22)(11,25)(13,26)(14,27)(17,29)(18,30)(19,28)(21,31)(24,32);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(25,32)(27,30);(3,13)(7,21)(11,24)(12,26)(14,18)(20,31)(25,32)(27,30)Polycyclic presentation:
2^9|a^2=g,f^2=i,b^a=bf,f^a=fh,h^a=hi,e^b=ei,f^b=fi,g^b=ghi,d^c=di,g^f=gi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72234840, #72238719, #72238723, … |
| maximal normal subgroup of | #72234840, #72238719, #72238723, … |
| maximal quotient of | #72234836, #72249348, #72255791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529663);