#8118870

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3922 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)(17,18,23,22)(19,26,24,20)(21,29,30,25)(27,28,31,32)(33,34,40,36)(35,37,42,39)(38,41,47,44)(43,45,48,46);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,19,23,24)(18,20,22,26)(21,27,30,31)(25,28,29,32)(33,35)(34,39)(36,37)(38,43)(40,42)(41,46)(44,45)(47,48);(1,3,6,4)(2,7,9,10)(5,11,13,12)(8,14,15,16)(17,20,23,26)(18,24,22,19)(21,28,30,32)(25,31,29,27)(33,36,40,34)(35,37,42,39)(38,41,47,44)(43,46,48,45);(1,4,6,3)(2,7,9,10)(5,12,13,11)(8,14,15,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,37,40,39)(34,35,36,42)(38,45,47,46)(41,43,44,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,21,23,30)(18,25,22,29)(19,27,24,31)(20,28,26,32)(33,38,40,47)(34,41,36,44)(35,43,42,48)(37,45,39,46);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,22,23,18)(19,26,24,20)(21,29,30,25)(27,32,31,28)(33,39,40,37)(34,42,36,35)(38,46,47,45)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=gi,e^2=g,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, 8026066);