#8718419

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3929 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8625615);