#8464234

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=h,e^2=gi,f^2=gi,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, 8371430);