#8159089

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	#56839	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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:

(1,2,7,8)(3,4,5,9)(6,14,15,10)(11,16,13,12)(17,18)(19,28)(20,21)(22,29)(23,27)(24,32)(25,31)(26,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(2,8)(4,9)(10,14)(12,16)(17,19)(18,24)(20,29)(21,30)(22,31)(23,32)(25,26)(27,28)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,3,7,5)(2,9,8,4)(6,11,15,13)(10,16,14,12)(17,20,23,25)(18,21,27,31)(19,29,32,26)(22,24,30,28)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,4,7,9)(2,3,8,5)(6,12,15,16)(10,11,14,13)(17,21,23,31)(18,25,27,20)(19,30,32,22)(24,26,28,29)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,5,7,3)(2,4,8,9)(6,13,15,11)(10,12,14,16)(17,18)(19,24)(20,21)(22,26)(23,27)(25,31)(28,32)(29,30);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,22,23,30)(18,26,27,29)(19,31,32,21)(20,24,25,28)(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=i,c^2=gi,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, 8066285);