#8750873

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	#53116	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4127 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=gi,e^2=g,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, 8658069);