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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(6,9)(7,11)(13,14,19,25)(15,26,30,21)(16,23,17,32)(18,24,20,35)(22,33,40,31)(27,42,28,37)(29,39,34,38)(36,44,41,43);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15)(14,21)(16,27)(17,28)(18,34)(19,30)(20,29)(22,36)(23,37)(24,39)(25,26)(31,44)(32,42)(33,43)(35,38)(40,41);(2,4)(13,16,19,17)(14,22,25,40)(15,27,30,28)(18,31,20,33)(21,36,26,41)(23,24,32,35)(29,43,34,44)(37,39,42,38);(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,23,25,32)(15,28,30,27)(18,31,20,33)(21,37,26,42)(22,24,40,35)(29,43,34,44)(36,39,41,38);(2,4)(6,9)(10,12)(13,18,19,20)(14,24,25,35)(15,29,30,34)(16,31,17,33)(21,38,26,39)(22,32,40,23)(27,43,28,44)(36,37,41,42)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=f,e^2=i,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, 6579013);