#8418678

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=h,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, 8325874);