#8157085

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

4774 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)(10,14)(11,17)(16,22)(18,19)(25,26,30,34)(27,32,35,31)(28,38,33,29)(36,37,39,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(27,35)(31,32)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(11,17)(15,21)(18,19)(23,24)(25,27,30,35)(26,31,34,32)(28,36,33,39)(29,37,38,40);(2,5)(6,7)(9,11)(10,15)(12,19)(13,17)(14,21)(16,24)(18,20)(22,23)(25,27)(26,32)(28,39)(29,37)(30,35)(31,34)(33,36)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,13,20)(10,16,14,22)(11,18,17,19)(15,23,21,24)(25,28,30,33)(26,29,34,38)(27,36,35,39)(31,37,32,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,29)(26,33)(27,37)(28,34)(30,38)(31,39)(32,36)(35,40)

Polycyclic presentation:

2^9|a^2=h,c^2=i,e^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, 8064281);