#9064086

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	#45319	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4335 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8971282);