#8118584

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	#57135	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8025780);