#8267608

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	#37317	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4398 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=hi,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, 8174804);