#8159284

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3859 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,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, 8066480);