#8750131

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×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3813 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=hi,d^2=g,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, 8657327);