#8750772

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(18,20)(23,25)(24,28)(31,32)(33,34,39,38)(35,42,36,40)(37,41,45,46)(43,48,44,47);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,18,22,20)(19,25,27,23)(21,24,30,28)(26,32,29,31)(33,35,39,36)(34,40,38,42)(37,43,45,44)(41,47,46,48);(2,3)(7,9)(8,10)(15,16)(17,18,22,20)(19,25,27,23)(21,24,30,28)(26,32,29,31)(33,34,39,38)(35,40,36,42)(37,41,45,46)(43,47,44,48);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,19,22,27)(18,23,20,25)(21,29,30,26)(24,32,28,31)(33,36,39,35)(34,40,38,42)(37,44,45,43)(41,47,46,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20,22,18)(19,25,27,23)(21,24,30,28)(26,31,29,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,45)(40,47)(42,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,38,39,34)(35,42,36,40)(37,41,45,46)(43,47,44,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=gi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657968);