#8750081

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	#2414	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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=g,e^2=h,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, 8657277);