#8256619

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	#56353	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4466 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8163815);