#8718001

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	#43105	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4524 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8625197);