#8231463

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4163 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^2=g,f^2=gi,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, 8138659);