#8206148

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4362 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^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, 8113344);