Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2059 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,23,19,24,17,20)(15,25,28,35,26,21,34,37)(16,29,30,36,33,22,32,40)(27,38,42,44,31,39,41,43);(5,6)(7,11)(8,10)(9,12)(14,20)(15,26)(16,30)(17,18)(21,37)(23,24)(25,35)(27,41)(31,42)(32,33)(36,40)(38,39);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15,19,26)(14,21,24,25)(16,31,33,27)(17,34,18,28)(20,35,23,37)(22,39,29,38)(30,41,32,42)(36,44,40,43);(7,12)(9,11)(13,16,17,32,19,33,18,30)(14,22,20,36,24,29,23,40)(15,27,34,41,26,31,28,42)(21,38,35,43,25,39,37,44);(6,10)(9,11)(13,17,19,18)(14,23,24,20)(15,28,26,34)(16,32,33,30)(21,35,25,37)(22,40,29,36)(27,42,31,41)(38,43,39,44)Polycyclic presentation:
2^9|a^2=fgi,c^2=i,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6914218);