#8691826

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	#44086	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4130 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8599022);