#8677135

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	#45522	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4322 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=g,f^2=gi,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, 8584331);