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 | |
7734 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)(13,14,18,19)(15,20,25,28)(16,21,26,24)(17,22,27,23);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12);(6,10)(7,11)(14,19)(17,27)(20,28)(21,24);(7,11)(8,12)(13,15)(14,20)(16,22)(17,24)(18,25)(19,28)(21,27)(23,26);(5,7)(6,8)(9,11)(10,12)(13,16)(14,21)(15,23)(17,20)(18,26)(19,24)(22,25)(27,28);(5,8)(6,7)(9,12)(10,11)(13,17)(14,22)(15,24)(16,20)(18,27)(19,23)(21,25)(26,28)Polycyclic presentation:
2^9|a^2=h,b^a=bg,c^a=ch,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72236848, #72238218, #72238241, … |
| maximal normal subgroup of | #72236848, #72238218, #72238241, … |
| maximal quotient of | #72333494, #72333567, #72333568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7966112);