#8159293

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	#53134	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4254 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,e^2=gh,f^2=gh,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, 8066489);