#8460720

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	#49550	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4518 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)(4,6)(5,8)(9,10,16,14)(11,20,22,17)(12,23,18,13)(15,19,24,21)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11,16,22)(10,17,14,20)(12,15,18,24)(13,19,23,21)(25,27,29,35)(26,30,33,31)(28,34,37,40)(32,38,36,39);(9,12)(10,13)(11,15)(14,23)(16,18)(17,19)(20,21)(22,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(1,2,4,6)(3,5,8,7)(9,13,16,23)(10,18,14,12)(11,21,22,19)(15,20,24,17)(25,26)(27,31)(28,36)(29,33)(30,35)(32,37)(34,38)(39,40);(1,3)(2,5)(4,8)(6,7)(9,14,16,10)(11,20,22,17)(12,23,18,13)(15,21,24,19);(1,2)(3,5)(4,6)(7,8)(9,15,16,24)(10,19,14,21)(11,18,22,12)(13,17,23,20)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,36)(35,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=g,e^2=hi,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, 8367916);