#8323219

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	#40044	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4398 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,e^2=g,f^2=gi,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, 8230415);