#8371344

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	#45518	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4432 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,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, 8278540);