#8393254

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	#35692	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4613 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300450);