#8467217

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	#37346	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4393 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)(3,4)(5,7)(6,8)(9,10,16,14)(11,21,23,17)(12,19,13,18)(15,20,22,24)(25,26,31,30)(27,32,37,35)(28,33,38,40)(29,39,36,34);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,22)(13,15)(14,21)(16,23)(18,24)(19,20)(25,26,31,30)(27,35,37,32)(28,33,38,40)(29,34,36,39);(9,12,16,13)(10,18,14,19)(11,22,23,15)(17,24,21,20)(25,27,31,37)(26,32,30,35)(28,36,38,29)(33,34,40,39);(2,7)(4,6)(9,13,16,12)(10,19,14,18)(11,22,23,15)(17,24,21,20)(25,28)(26,33)(27,36)(29,37)(30,40)(31,38)(32,34)(35,39);(1,3)(2,6)(4,7)(5,8)(9,14,16,10)(11,21,23,17)(12,19,13,18)(15,24,22,20)(25,29,31,36)(26,34,30,39)(27,28,37,38)(32,40,35,33);(1,4,5,6)(2,3,7,8)(9,15,16,22)(10,20,14,24)(11,13,23,12)(17,19,21,18)(25,30,31,26)(27,35,37,32)(28,33,38,40)(29,39,36,34)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8374413);