#8159287

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3852 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)(3,4,5,8)(6,14,15,10)(11,16,13,12)(17,18,23,22)(19,24,30,29)(20,26,21,25)(27,32,28,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3,7,5)(2,8,9,4)(6,11,15,13)(10,16,14,12)(17,19,23,30)(18,24,22,29)(20,27,21,28)(25,31,26,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(2,9)(4,8)(10,14)(12,16)(17,20,23,21)(18,25,22,26)(19,27,30,28)(24,31,29,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,4,7,8)(2,3,9,5)(6,12,15,16)(10,11,14,13)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,5,7,3)(2,4,9,8)(6,13,15,11)(10,12,14,16)(17,21,23,20)(18,26,22,25)(19,28,30,27)(24,32,29,31);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,16)(9,14)(17,22,23,18)(19,29,30,24)(20,26,21,25)(27,32,28,31)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066483);