#8783231

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	#45572	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3903 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8690427);