Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622911 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C25 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
3712 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)(4,18)(5,12)(6,13)(7,23)(8,25)(9,14)(10,16)(15,29)(17,27)(19,26)(20,32)(21,30)(22,31)(24,28);(1,3)(2,10)(4,8)(5,15)(6,16)(7,24)(9,17)(11,23)(12,26)(13,27)(14,28)(18,31)(19,22)(20,21)(25,32)(29,30);(1,4)(2,11)(3,5)(6,19)(7,20)(8,17)(9,15)(10,13)(12,29)(14,27)(16,21)(18,25)(22,24)(23,28)(26,31)(30,32);(1,5,9,8)(2,12,14,25)(3,15,17,4)(6,22,7,21)(10,26,28,32)(11,29,27,18)(13,31,23,30)(16,19,24,20);(1,6,9,7)(2,13,14,23)(3,16,17,24)(4,19,15,20)(5,21,8,22)(10,27,28,11)(12,30,25,31)(18,26,29,32)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ci,f^b=fi,g^b=ghi,f^c=fh,g^c=gi,e^d=ei,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72260522, #72260545, #72337673, … |
| maximal normal subgroup of | #72260522, #72260545, #72337673, … |
| maximal quotient of | #72261184, #72262727, #72337680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530109);