#8159139

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	#42752	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3929 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(2,4)(7,10)(8,12)(14,16)(17,18)(19,25)(20,23)(21,24)(22,28)(26,30)(27,31)(29,32)(34,36)(39,41)(40,44)(46,47);(3,9)(7,10)(11,15)(14,16)(18,20)(24,25)(26,28)(31,32)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(1,2,6,4)(3,7,9,10)(5,8,13,12)(11,14,15,16)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,4,6,2)(3,10,9,7)(5,12,13,8)(11,16,15,14)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=g,e^2=h,f^2=gh,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^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, 8066335);