#8115548

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	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022744);