#8463667

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	#45356	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4384 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=g,e^2=ghi,f^2=hi,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, 8370863);