#8159140

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	#2431	C₂²
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)(17,18,23,24)(19,20,21,25)(22,30,31,26)(27,32,29,28)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(3,9)(7,10)(11,15)(14,16)(18,24)(20,25)(26,30)(28,32)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,2,6,4)(3,7,9,10)(5,8,13,12)(11,14,15,16)(17,19,23,21)(18,25,24,20)(22,27,31,29)(26,32,30,28)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20,23,25)(18,19,24,21)(22,28,31,32)(26,27,30,29)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,4,6,2)(3,10,9,7)(5,12,13,8)(11,16,15,14)(17,21,23,19)(18,20,24,25)(22,29,31,27)(26,28,30,32);(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)(23,31)(24,30)(25,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=g,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, 8066336);