#8751287

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3900 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)(3,10,5,8)(4,9,12,14)(11,16,13,15)(17,18)(19,28)(20,26)(21,29)(22,24)(23,25)(27,31)(30,32)(34,36)(39,41)(40,44)(46,47);(1,3,7,5)(2,8,6,10)(4,11,12,13)(9,15,14,16)(17,19,23,22)(18,24,25,28)(20,29,30,27)(21,26,31,32)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(4,12)(9,14)(11,13)(15,16)(18,25)(20,30)(24,28)(27,29)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,20)(18,26)(19,27)(21,24)(22,29)(23,30)(25,32)(28,31)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,5,7,3)(2,8,6,10)(4,13,12,11)(9,15,14,16)(17,21)(18,27)(19,26)(20,24)(22,32)(23,31)(25,29)(28,30)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47);(1,6,7,2)(3,10,5,8)(4,9,12,14)(11,15,13,16)(17,22,23,19)(18,24,25,28)(20,27,30,29)(21,26,31,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=i,e^2=gh,f^2=gi,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, 8658483);