#8437728

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	#56578	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4355 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)(4,6)(5,8)(10,15)(11,18)(17,22)(19,24)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,3)(2,5)(4,8)(6,7)(9,10,14,15)(11,13,18,16)(12,17,21,22)(19,20,24,23)(27,38)(28,39)(32,37)(33,40);(9,11)(10,13)(12,19)(14,18)(15,16)(17,20)(21,24)(22,23)(25,27,31,38)(26,32,36,37)(28,35,39,29)(30,33,34,40);(2,6)(5,7)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,22)(21,24)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,34)(36,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,21)(10,17,15,22)(11,19,18,24)(13,20,16,23)(25,29,31,35)(26,34,36,30)(27,28,38,39)(32,40,37,33);(1,2)(3,5)(4,6)(7,8)(9,13,14,16)(10,18,15,11)(12,20,21,23)(17,24,22,19)(25,30,31,34)(26,35,36,29)(27,33,38,40)(28,32,39,37)

Polycyclic presentation:

2^9|b^2=hi,c^2=i,e^2=h,f^2=h,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, 8344924);