#8127449

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

4336 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:

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

Polycyclic presentation:

2^9|a^2=g,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, 8034645);