Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667508 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24196 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10,12,13)(11,14,15,16)(17,18,20,23)(19,25,28,22)(21,24,29,32)(26,30,27,31);(1,2)(3,6)(4,5)(7,8)(17,19)(18,22)(20,28)(21,26)(23,25)(24,31)(27,29)(30,32);(10,13)(14,16)(18,23)(19,26)(21,29)(22,30)(25,31)(27,28);(1,3,4,7)(2,6,5,8)(18,23)(22,25)(24,32)(30,31);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(18,23)(19,27)(21,29)(22,31)(25,30)(26,28)Polycyclic presentation:
2^9|a^2=g,d^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72251197, #72256852, #72256902, … |
| maximal normal subgroup of | #72251197, #72256852, #72256902, … |
| maximal quotient of | #72265576 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6621769);