#8445853

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4258 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8353049);