#8119925

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4249 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^2=gi,f^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, 8027121);