#8807457

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	#39454	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4323 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8714653);