Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465212 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16516 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(4,15)(5,18)(8,23)(9,25)(10,22)(11,20)(13,19)(16,30)(21,27)(24,26)(28,32);(1,2,5,7,6,10,18,22)(3,11,19,23,14,8,13,20)(4,9,17,21,15,27,31,25)(12,28,30,26,29,24,16,32);(1,3)(2,8,10,11)(4,16)(5,19)(6,14)(7,20,22,23)(9,26,27,32)(12,17)(13,18)(15,30)(21,24,25,28)(29,31);(1,4,5,17,6,15,18,31)(2,9,7,21,10,27,22,25)(3,12,13,16,14,29,19,30)(8,24,23,26,11,28,20,32)Polycyclic presentation:
2^9|b^2=eg,c^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251783, #72251792, #72251801, … |
| maximal normal subgroup of | #72251783, #72251792, #72251801, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372408);