#8751123

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3877 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658319);