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 | |
7807 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)(6,10)(7,12)(13,14,18,19)(15,16,21,24)(17,20,25,26)(22,23,27,28);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12);(5,7)(6,11)(8,10)(9,12)(14,19)(15,21)(20,26)(22,27);(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,22)(18,21)(19,24)(20,23)(25,27)(26,28);(5,6)(7,11)(8,12)(9,10)(13,16)(14,15)(17,23)(18,24)(19,21)(20,22)(25,28)(26,27);(5,8,9,11)(6,7,10,12)(13,17,18,25)(14,20,19,26)(15,22,21,27)(16,23,24,28)Polycyclic presentation:
2^9|a^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72238174, #72238186, #72238187, … |
| maximal normal subgroup of | #72238174, #72238186, #72238187, … |
| maximal quotient of | #72326387, #72326400, #72327262, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7825017);