#7655503

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4308 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,4,6)(3,7,8,5)(26,31)(27,33)(34,39)(35,40);(1,3,4,8)(2,5,6,7)(9,10)(11,16)(12,15)(13,17)(14,19)(18,22)(20,23)(21,24);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,37)(36,40)(38,39);(9,12,13,20)(10,15,17,23)(11,18,19,24)(14,21,16,22)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,37,39,36);(9,11)(10,16)(12,18)(13,19)(14,17)(15,22)(20,24)(21,23)(25,28,30,32)(26,33,31,27)(29,37,38,36)(34,35,39,40);(9,10)(11,14)(12,15)(13,17)(16,19)(18,21)(20,23)(22,24)(25,29,30,38)(26,34,31,39)(27,35,33,40)(28,36,32,37)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7562699);