Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622451 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C23×#99 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C2×D4 | C25 |
| prime radical | #62842 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
4325 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)(4,8)(5,17,18,9)(6,10)(7,14,20,27)(12,23)(13,24)(15,21)(16,29,28,22)(19,26,30,32)(25,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,19)(9,11)(10,20)(15,25)(16,26)(17,24)(18,27)(21,30)(22,23)(28,32)(29,31);(1,4)(2,8)(3,12)(5,16)(6,15)(7,19)(9,22)(10,21)(11,23)(13,25)(14,26)(17,29)(18,28)(20,30)(24,31)(27,32);(4,15)(8,21)(12,25)(16,28)(19,30)(22,29)(23,31)(26,32);(3,13)(7,20)(11,24)(12,25)(14,27)(19,30)(23,31)(26,32)Polycyclic presentation:
2^9|a^2=g,b^a=bf,f^a=fh,h^a=hi,e^b=ei,g^b=gh,d^c=di,g^f=gi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72234840, #72238716, #72238720, … |
| maximal normal subgroup of | #72234840, #72238716, #72238720, … |
| maximal quotient of | #72234834, #72240218, #72240225, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529647);