Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493657 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20609 |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #63035 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,15,19,10)(4,20,18,30)(5,12,6,11)(7,13)(8,14)(9,17)(16,31)(21,32)(22,26)(23,29,25,28)(24,27);(1,3)(2,9)(4,7)(5,17)(6,24)(8,19)(10,12)(11,15)(13,26)(14,27)(16,18)(20,29)(21,23)(22,25)(28,30)(31,32);(1,4)(2,10)(3,16)(5,21)(6,22)(7,19)(8,18)(9,11)(12,27)(13,30)(14,15)(17,25)(20,31)(23,24)(26,29)(28,32);(1,5,8,6)(2,11,14,12)(3,17,19,24)(4,21,18,22)(7,23,16,25)(9,15,27,10)(13,28,31,29)(20,26,30,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,c^b=ci,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,g^d=gi,f^e=fi,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72260467, #72260522, #72260534, … |
| maximal normal subgroup of | #72260467, #72260522, #72260534, … |
| maximal quotient of | #72261064, #72261177, #72262723 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400854);