#8678980

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

3959 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8586176);