#8963123

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=gi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8870319);