#8093351

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

4402 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000547);