#8942343

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	#48168	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4130 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=h,e^2=gh,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, 8849539);