#8963438

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	#52048	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=ghi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=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, 8870634);