#7655172

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	#42024	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=g,e^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, 7562368);