#8436227

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	#37529	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8343423);