#8750285

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	#45063	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4307 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=gh,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, 8657481);