#8436567

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4365 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=h,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, 8343763);