#8751696

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4159 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658892);