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 | C2×#3277 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2111 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)(13,14,19,24)(15,25,28,21)(16,22,31,38)(17,23,18,20)(26,39,30,36)(27,37,32,34)(29,40,33,35)(41,43,42,44);(5,6)(7,11)(8,10)(9,12)(14,20)(16,29)(17,18)(21,34)(22,38)(23,24)(25,37)(26,41)(27,32)(30,42)(31,33)(36,39);(2,4)(5,7,8,12)(6,9,10,11)(13,15,19,28)(14,21,24,25)(16,30,31,26)(17,32,18,27)(20,34,23,37)(22,39,38,36)(29,42,33,41)(35,44,40,43);(7,12)(9,11)(13,16,19,31)(14,22,24,38)(15,26,28,30)(17,29,18,33)(20,35,23,40)(21,36,25,39)(27,42,32,41)(34,43,37,44);(6,10)(9,11)(13,17,19,18)(14,23,24,20)(15,27,28,32)(16,29,31,33)(21,34,25,37)(22,40,38,35)(26,42,30,41)(36,43,39,44)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6961838);