#8125651

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

4430 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=g,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^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, 8032847);