#8358520

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	#36361	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8265716);