#8125478

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	#52696	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4081 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)(3,7,12,10)(4,9,14,16)(5,13,15,11)(17,18,23,20)(19,24,26,21)(22,28,30,25)(27,29,32,31)(34,37)(35,36)(42,45)(43,44);(1,3)(2,7)(4,11)(5,9)(6,12)(8,10)(13,14)(15,16)(17,19,23,26)(18,24,20,21)(22,27,30,32)(25,31,28,29)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(2,8)(3,12)(9,16)(11,13)(19,26)(21,24)(27,32)(29,31)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(1,4,6,14)(2,9,8,16)(3,13,12,11)(5,7,15,10)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,3)(2,10)(4,13)(5,9)(6,12)(7,8)(11,14)(15,16)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,37,39,34)(35,41,36,40)(38,45,46,42)(43,48,44,47);(1,5)(2,11)(3,9)(4,7)(6,15)(8,13)(10,14)(12,16)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=i,e^2=g,f^2=ghi,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, 8032674);