#8429461

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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)(3,11)(4,8)(5,6)(7,10)(9,16)(12,15)(13,14)(17,18,23,26)(19,21,24,20)(22,25,31,30)(27,29,32,28)(33,34,40,39)(35,36,42,38)(37,45,48,41)(43,47,46,44);(3,4)(8,11)(12,14)(13,15)(17,19,23,24)(18,20,26,21)(22,27,31,32)(25,28,30,29)(33,35)(34,38)(36,39)(37,46)(40,42)(41,44)(43,48)(45,47);(1,3,7,4)(2,8,10,11)(5,12,16,14)(6,13,9,15)(17,20,23,21)(18,19,26,24)(22,28,31,29)(25,27,30,32)(33,36,40,38)(34,35,39,42)(37,44,48,47)(41,43,45,46);(1,4,7,3)(2,8,10,11)(5,14,16,12)(6,13,9,15)(17,18)(19,20)(21,24)(22,30)(23,26)(25,31)(27,29)(28,32)(33,37,40,48)(34,41,39,45)(35,43,42,46)(36,44,38,47);(1,5)(2,9)(3,12)(4,14)(6,10)(7,16)(8,15)(11,13)(17,21,23,20)(18,24,26,19)(22,29,31,28)(25,32,30,27)(33,38,40,36)(34,42,39,35)(37,47,48,44)(41,46,45,43);(1,6,7,9)(2,5,10,16)(3,13,4,15)(8,12,11,14)(17,22,23,31)(18,25,26,30)(19,27,24,32)(20,28,21,29)(33,39,40,34)(35,36,42,38)(37,41,48,45)(43,47,46,44)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8336657);