#8889050

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	#46807	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4211 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)(3,5,10,8)(4,12,13,9)(11,16,14,15)(18,20)(19,25)(24,28)(26,31)(35,41)(38,42)(43,47)(46,48);(2,7)(4,13)(5,8)(11,14)(17,18,22,20)(19,23,25,27)(21,28,29,24)(26,32,31,30)(33,34)(35,38)(36,44)(37,40)(39,45)(41,42)(43,48)(46,47);(2,7)(3,10)(9,12)(11,14)(19,25)(23,27)(26,31)(30,32)(33,34,37,40)(35,38,41,42)(36,39,45,44)(43,46,47,48);(1,3)(2,8)(4,14)(5,7)(6,10)(9,15)(11,13)(12,16)(17,19,22,25)(18,23,20,27)(21,26,29,31)(24,30,28,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,45)(46,47);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32)(33,35,37,41)(34,38,40,42)(36,43,45,47)(39,46,44,48);(1,5,6,8)(2,3,7,10)(4,15,13,16)(9,11,12,14)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,36,37,45)(34,39,40,44)(35,43,41,47)(38,46,42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=gh,e^2=gi,f^2=hi,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, 8796246);