#8418442

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	#36670	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3867 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8325638);