#8326055

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8233251);