#8876190

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3900 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=h,f^2=i,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, 8783386);