#8126003

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4256 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)(9,10,14,18)(11,19,21,15)(12,22,20,16)(13,17,23,24)(27,33)(30,36)(34,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,18,19)(12,13,20,23)(16,17,22,24)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(2,5)(6,8)(12,20)(13,23)(16,22)(17,24)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,20)(10,16,18,22)(11,13,21,23)(15,17,19,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,20)(12,21)(14,23)(15,22)(16,19)(18,24)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,22)(23,24)(25,28,29,35)(26,31,32,38)(27,34,33,39)(30,37,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,f^2=h,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, 8033199);