#8119897

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	#46983	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8027093);