#8689860

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	#45563	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4588 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,e^2=gh,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, 8597056);