#8658129

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	#45444	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4711 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8565325);