Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2814 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)(7,11)(8,10)(9,12)(13,14)(15,19)(16,27)(17,30)(18,22)(20,26)(21,28)(23,33)(24,37)(25,36)(29,39)(31,35)(32,38)(34,41)(40,44)(42,43);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,24)(17,26)(18,25)(20,30)(21,34)(22,36)(23,35)(27,37)(28,41)(29,40)(31,33)(32,42)(38,43)(39,44);(2,4)(13,15)(14,20)(16,24)(17,31)(18,25)(19,30)(21,37)(22,35)(23,36)(26,33)(27,34)(28,42)(29,40)(32,41)(38,44)(39,43);(6,10)(7,12)(13,16,18,29)(14,21,22,38)(15,24,25,40)(17,32,33,28)(19,34,36,43)(20,37,35,44)(23,39,30,27)(26,42,31,41);(14,22)(15,25)(21,38)(23,30)(24,40)(26,31)(27,39)(41,42)Polycyclic presentation:
2^9|d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407919);