#9390431

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	#46544	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3898 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=hi,d^2=gh,e^2=gh,f^2=gh,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=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9297627);