#8436565

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

4392 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)(9,10,14,18)(11,15,16,12)(13,17,22,21)(19,23,24,20)(25,26,32,30)(27,36,37,33)(28,38,34,29)(31,35,39,40);(1,3)(2,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,24)(21,22)(25,27,32,37)(26,33,30,36)(28,35,34,40)(29,39,38,31);(1,4)(2,3)(5,8)(6,7)(9,11,14,16)(10,15,18,12)(13,19,22,24)(17,23,21,20)(25,28)(26,29)(27,35)(30,38)(31,36)(32,34)(33,39)(37,40);(1,5,6,3)(2,4,8,7)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,20)(23,24)(25,29,32,38)(26,34,30,28)(27,31,37,39)(33,35,36,40);(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,24,21,19)(25,30,32,26)(27,36,37,33)(28,38,34,29)(31,40,39,35);(1,4)(2,5)(3,8)(6,7)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,31)(26,35)(27,29)(28,36)(30,40)(32,39)(33,34)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,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, 8343761);