Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| 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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,20,28,19,23,18,17)(15,22,27,21,16,26,25,24);(5,6)(7,11)(8,10)(9,12)(13,15)(14,21)(16,19)(17,26)(18,27)(20,25)(22,28)(23,24);(2,4)(5,7,8,12)(6,9,10,11)(13,16,19,15)(14,22,23,26)(17,24,28,21)(18,27,20,25);(7,12)(9,11)(13,17,20,14,19,28,18,23)(15,24,27,22,16,21,25,26);(6,10)(9,11)(15,16)(21,24)(22,26)(25,27)Polycyclic presentation:
2^9|a^2=fg,c^2=h,d^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285261, #72307869, #72308011, … |
| maximal normal subgroup of | #72285261, #72307869, #72308011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972333);