#8172654

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	#59050	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4366 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)(9,10,15,12)(11,16,18,14)(13,20,22,17)(19,21,24,23)(25,26)(27,33)(28,29)(30,38)(31,32)(34,39)(35,36)(37,40);(1,3,6,5)(2,7,4,8)(9,11,15,18)(10,16,12,14)(13,19,22,24)(17,23,20,21)(26,32)(28,33)(34,38)(36,40);(1,3,6,5)(2,8,4,7)(10,12)(11,18)(17,20)(19,24)(25,27,31,29)(26,28,32,33)(30,35,39,37)(34,36,38,40);(1,4,6,2)(3,7,5,8)(9,12,15,10)(11,16,18,14)(13,17,22,20)(19,21,24,23)(25,28)(26,27)(29,32)(30,36)(31,33)(34,35)(37,38)(39,40);(1,5,6,3)(2,7,4,8)(9,13,15,22)(10,17,12,20)(11,19,18,24)(14,21,16,23)(25,29,31,27)(26,33,32,28)(30,37,39,35)(34,40,38,36);(9,14,15,16)(10,18,12,11)(13,21,22,23)(17,24,20,19)(25,30,31,39)(26,34,32,38)(27,35,29,37)(28,36,33,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8079850);