#8152156

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	#45303	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4283 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8059352);