#8359930

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	#42079	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4078 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,16)(14,15)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(33,34)(35,39)(36,44)(37,38)(40,46)(41,43)(42,48)(45,47);(2,7)(3,10)(4,13)(15,16)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,34)(35,41)(36,44)(37,38)(39,43)(40,46)(42,47)(45,48);(2,7)(5,8)(9,12)(15,16)(22,30)(26,29)(27,28)(31,32)(34,38)(39,41)(40,44)(47,48);(1,3)(2,8)(4,14)(5,7)(6,10)(9,15)(11,13)(12,16)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,46)(44,47);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,21,23,18)(19,24,20,25)(22,29,30,26)(27,31,28,32)(33,36)(34,40)(35,42)(37,46)(38,44)(39,47)(41,48)(43,45);(1,5,6,8)(2,3,7,10)(4,15,13,16)(9,11,12,14)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8267126);