#8158962

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4251 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=g,f^2=gh,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^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, 8066158);