#8534533

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3814 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8441729);