#8159059

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	#57574	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 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)(3,7)(4,5)(6,8)(9,10,15,18)(11,12,13,16)(14,22,23,17)(19,24,21,20)(25,26)(27,36)(28,29)(30,39)(31,35)(32,38)(33,37)(34,40);(2,5)(6,7)(9,11,15,13)(10,16,18,12)(14,19,23,21)(17,24,22,20)(25,27,31,38)(26,32,35,36)(28,34,33,39)(29,30,37,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,16,18,12)(14,19,23,21)(17,24,22,20)(25,28,31,33)(26,29,35,37)(27,34,38,39)(30,36,40,32);(1,3,4,8)(2,6,5,7)(9,12,15,16)(10,11,18,13)(14,20,23,24)(17,19,22,21)(25,29,31,37)(26,33,35,28)(27,30,38,40)(32,39,36,34);(9,13,15,11)(10,12,18,16)(14,21,23,19)(17,20,22,24)(25,26)(27,32)(28,29)(30,34)(31,35)(33,37)(36,38)(39,40);(9,14,15,23)(10,17,18,22)(11,19,13,21)(12,20,16,24)(25,30)(26,34)(27,37)(28,36)(29,38)(31,40)(32,33)(35,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gi,d^2=g,e^2=h,f^2=h,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, 8066255);