#8839170

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	#45462	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8746366);