#8119207

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	#39380	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4330 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,15)(13,21)(14,18)(17,22)(19,23)(20,24)(26,30)(28,35)(31,36)(34,39);(1,2,4,6)(3,5,8,7)(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(27,33)(31,36)(34,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,12)(10,16,18,15)(13,19,22,20)(17,24,21,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,11)(10,16,18,15)(13,20,22,19)(17,24,21,23)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,24)(21,22)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026403);