#8436571

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4267 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)(3,7)(4,5)(6,8)(9,10,15,18)(11,17,13,16)(12,21,23,14)(19,20,22,24)(25,26,31,36)(27,30,35,32)(28,29,34,33)(37,39,40,38);(1,3)(2,7)(4,8)(5,6)(9,11,15,13)(10,16,18,17)(12,19,23,22)(14,20,21,24)(25,27)(26,32)(28,37)(29,38)(30,36)(31,35)(33,39)(34,40);(1,4)(2,3)(5,8)(6,7)(25,28,31,34)(26,33,36,29)(27,37,35,40)(30,38,32,39);(1,5,6,3)(2,4,8,7)(9,12,15,23)(10,14,18,21)(11,19,13,22)(16,20,17,24)(25,26)(27,30)(28,33)(29,34)(31,36)(32,35)(37,38)(39,40);(9,13,15,11)(10,17,18,16)(12,22,23,19)(14,24,21,20)(25,29,31,33)(26,34,36,28)(27,38,35,39)(30,37,32,40);(1,4)(2,5)(3,8)(6,7)(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,22)(18,23)(25,30)(26,35)(27,36)(28,38)(29,37)(31,32)(33,40)(34,39)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=hi,d^2=gh,e^2=g,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, 8343767);