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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1823 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)(15,26,33,37,30,20,29,41)(16,22,34,21,32,40,31,38)(27,36,43,42,28,39,44,35);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,30)(14,20,24,26)(16,27,32,28)(17,33,19,29)(21,35,38,42)(22,39,40,36)(23,41,25,37)(31,44,34,43);(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,27)(17,31)(18,32)(19,34)(20,35)(22,23)(24,38)(25,40)(26,42)(28,30)(29,43)(33,44)(36,37)(39,41);(1,3,4,2)(7,12)(9,11)(13,16)(14,22)(15,28)(17,34)(18,32)(19,31)(20,36)(21,23)(24,40)(25,38)(26,39)(27,30)(29,43)(33,44)(35,37)(41,42);(6,10)(9,11)(13,17,18,19)(14,23,24,25)(15,29,30,33)(16,31,32,34)(20,37,26,41)(21,22,38,40)(27,43,28,44)(35,36,42,39)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,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, 6971478);