Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,8)(9,11)(10,12)(13,14,20,25)(15,21,16,28)(17,31,34,41)(18,24,19,35)(22,27,37,30)(23,32,40,33)(26,42,29,38)(36,44,39,43);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,21)(16,20)(17,29)(18,30)(19,27)(22,24)(23,39)(25,28)(26,34)(31,38)(32,43)(33,44)(35,37)(36,40)(41,42);(2,4)(13,16,20,15)(14,22,25,37)(17,29,34,26)(18,30,19,27)(21,35,28,24)(23,38,40,42)(31,39,41,36)(32,44,33,43);(6,11)(7,12)(13,17,20,34)(14,23,25,40)(15,26,16,29)(18,33,19,32)(21,36,28,39)(22,38,37,42)(24,41,35,31)(27,44,30,43);(5,8,9,10)(6,7,11,12)(13,18,20,19)(14,24,25,35)(15,27,16,30)(17,32,34,33)(21,37,28,22)(23,31,40,41)(26,43,29,44)(36,38,39,42)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,e^b=ei,g^c=gi,e^d=ei,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410202);