#8118845

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3928 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=i,f^2=g,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, 8026041);