Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7623 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,21)(15,19,22,16)(17,20,26,28)(23,27,25,24);(1,2)(3,4)(13,15)(14,19)(16,21)(17,25)(18,22)(20,24)(23,26)(27,28);(6,9)(10,11)(15,22)(16,19)(23,25)(24,27);(5,6)(7,10)(8,9)(11,12)(13,16,18,19)(14,15,21,22)(17,24,26,27)(20,23,28,25);(5,7,8,12)(6,10,9,11)(13,15)(14,19)(16,21)(17,23)(18,22)(20,27)(24,28)(25,26);(13,17)(14,20)(15,23)(16,24)(18,26)(19,27)(21,28)(22,25)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72327640 |
| maximal normal subgroup of | #72327640 |
| maximal quotient of | #72327854, #72328062, #72328242, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7643372);