#8139539

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

4646 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)(6,8)(10,13)(11,17)(16,21)(18,20)(27,34)(29,37)(31,32)(33,36);(3,7)(6,8)(9,10,14,13)(11,15,17,19)(12,16,22,21)(18,23,20,24)(25,26)(27,32)(28,30)(29,33)(31,34)(35,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(25,27)(26,31)(28,32)(29,35)(30,34)(33,38)(36,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,20)(13,19)(14,17)(16,24)(18,22)(21,23)(25,28,30,26)(27,32,34,31)(29,36,37,33)(35,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,12,14,22)(10,16,13,21)(11,18,17,20)(15,23,19,24)(25,27)(26,32)(28,31)(29,35)(30,34)(33,39)(36,38)(37,40);(9,13,14,10)(11,19,17,15)(12,21,22,16)(18,24,20,23)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|b^2=g,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8046735);