Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3486 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,20,26)(15,27,31,39)(16,32,28,41)(17,23,37,38)(18,24,19,25)(21,30,40,29)(22,34,42,33)(35,43,36,44);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,20,31)(14,21,26,40)(16,17,28,37)(18,30,19,29)(22,38,42,23)(24,27,25,39)(32,44,41,43)(33,36,34,35);(7,11)(9,12)(13,16,20,28)(14,22,26,42)(15,17,31,37)(18,33,19,34)(21,38,40,23)(24,41,25,32)(27,43,39,44)(29,35,30,36);(7,11)(9,12)(13,17)(14,23)(15,28)(16,31)(18,36)(19,35)(20,37)(21,22)(24,44)(25,43)(26,38)(27,41)(29,33)(30,34)(32,39)(40,42);(6,10)(9,12)(13,18,20,19)(14,24,26,25)(15,29,31,30)(16,33,28,34)(17,35,37,36)(21,39,40,27)(22,41,42,32)(23,43,38,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377416);