#8317530

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	#52768	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3917 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)(3,11)(4,14)(5,10)(6,8)(7,13)(9,15)(12,16)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(1,3,6,13)(2,7,8,11)(4,12,15,10)(5,9,16,14)(17,18,22,23)(19,21,25,30)(20,24,29,31)(26,28,27,32)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(2,8)(5,16)(7,11)(9,14)(18,23)(21,30)(24,31)(28,32)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,11)(12,15)(14,16)(17,19,22,25)(18,21,23,30)(20,27,29,26)(24,32,31,28)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,4)(2,9)(3,12)(5,11)(6,15)(7,16)(8,14)(10,13)(17,20,22,29)(18,24,23,31)(19,26,25,27)(21,28,30,32)(33,34,39,38)(35,40,37,42)(36,41,46,45)(43,47,44,48);(1,5,6,16)(2,10,8,12)(3,9,13,14)(4,11,15,7)(17,21)(18,25)(19,23)(20,28)(22,30)(24,27)(26,31)(29,32)(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=gh,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8224726);