#8118854

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	#55149	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=gi,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, 8026050);