#8165185

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4450 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,e^2=hi,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, 8072381);