Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #45136 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,17,23,18,24,19,25)(15,26,34,38,30,20,29,41)(16,31,35,21,33,22,32,39)(27,40,43,42,28,37,44,36);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,30)(14,20,24,26)(16,27,33,28)(17,34,19,29)(21,36,39,42)(22,40,31,37)(23,41,25,38)(32,44,35,43);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,27)(17,32)(18,33)(19,35)(20,36)(22,25)(23,31)(24,39)(26,42)(28,30)(29,43)(34,44)(37,41)(38,40);(1,3,4,2)(7,12)(9,11)(13,16)(14,22)(15,28)(17,35)(18,33)(19,32)(20,37)(21,25)(23,39)(24,31)(26,40)(27,30)(29,43)(34,44)(36,41)(38,42);(6,10)(9,11)(13,17,18,19)(14,23,24,25)(15,29,30,34)(16,32,33,35)(20,38,26,41)(21,31,39,22)(27,43,28,44)(36,40,42,37)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6971734);