#8317529

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	#56366	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3915 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8224725);