Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2043 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)(7,10)(8,12)(9,11)(13,14,18,23)(15,19,24,32)(16,21,27,35)(17,20,29,22)(25,31,37,42)(26,33,38,30)(28,34,40,36)(39,43,44,41);(2,3)(5,7,10,6)(8,12,9,11)(13,15)(14,19)(16,25)(17,26)(18,24)(20,33)(21,31)(22,30)(23,32)(27,37)(28,39)(29,38)(34,43)(35,42)(36,41)(40,44);(1,2,4,3)(14,20)(15,24)(17,29)(19,30)(21,34)(22,23)(25,37)(28,40)(31,41)(32,33)(35,36)(42,43);(5,8,10,9)(6,11,7,12)(13,16)(14,21)(15,25)(17,28)(18,27)(19,31)(20,34)(22,36)(23,35)(24,37)(26,39)(29,40)(30,41)(32,42)(33,43)(38,44);(1,3,4,2)(5,9,10,8)(6,11,7,12)(15,24)(16,27)(19,32)(21,35)(26,38)(28,40)(30,33)(34,36)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=hi,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6718848);