#8689852

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4492 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8597048);