#8467211

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

4298 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)(3,8)(4,6)(5,7)(9,10,15,14)(11,16,21,19)(12,17,22,24)(13,23,20,18)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,37)(39,40);(1,3,5,4)(2,6,7,8)(9,10,15,14)(11,19,21,16)(12,17,22,24)(13,18,20,23)(26,28)(27,33)(32,36)(35,39);(9,11,15,21)(10,16,14,19)(12,20,22,13)(17,18,24,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(2,7)(6,8)(9,12)(10,17)(11,20)(13,21)(14,24)(15,22)(16,18)(19,23)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,4,5,3)(2,8,7,6)(9,13,15,20)(10,18,14,23)(11,12,21,22)(16,24,19,17)(25,29,31,38)(26,32,28,36)(27,35,33,39)(30,37,34,40);(1,2)(3,6)(4,8)(5,7)(9,14,15,10)(11,19,21,16)(12,17,22,24)(13,23,20,18)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=hi,d^2=h,e^2=ghi,f^2=i,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, 8374407);