#8127461

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4368 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)(3,5)(4,7)(6,8)(9,10)(11,13)(12,15)(14,20)(16,17)(18,22)(19,23)(21,24)(25,26,29,32)(27,30,34,39)(28,35,37,31)(33,40,36,38);(2,3)(6,7)(9,11,15,17)(10,16,12,13)(14,19,22,24)(18,23,20,21)(25,26)(27,30)(28,31)(29,32)(33,38)(34,39)(35,37)(36,40);(1,2,5,3)(4,6,8,7)(10,12)(13,16)(18,20)(21,23)(25,27,29,34)(26,30,32,39)(28,36,37,33)(31,40,35,38);(1,3,5,2)(4,7,8,6)(9,12,15,10)(11,16,17,13)(14,20,22,18)(19,23,24,21)(25,28,29,37)(26,31,32,35)(27,33,34,36)(30,38,39,40);(1,4,5,8)(2,6,3,7)(9,13,15,16)(10,17,12,11)(14,21,22,23)(18,24,20,19)(25,27)(26,30)(28,33)(29,34)(31,38)(32,39)(35,40)(36,37);(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,26)(27,30)(28,31)(29,32)(33,38)(34,39)(35,37)(36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8034657);