#8843852

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	#54877	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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:

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

Polycyclic presentation:

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

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, 8751048);