#8256593

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	#56839	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=gh,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8163789);