#8739269

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

3875 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)(3,11,5,8)(4,10,14,15)(6,12,16,13)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32)(34,36)(39,42)(40,44)(46,48);(1,3,7,5)(2,8,9,11)(4,13,14,12)(6,10,16,15)(18,23)(19,26)(25,28)(27,32)(35,41)(39,42)(43,47)(46,48);(2,9)(4,14)(8,11)(12,13)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(1,4)(2,10)(3,12)(5,13)(6,8)(7,14)(9,15)(11,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,5,7,3)(2,11,9,8)(4,13,14,12)(6,10,16,15)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,6,7,16)(2,12,9,13)(3,10,5,15)(4,11,14,8)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8646465);