#8260797

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	#36411	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4270 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gi,d^2=g,e^2=gi,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, 8167993);