#10040517

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

3754 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,15)(14,16)(17,18)(19,21)(20,25)(22,24)(23,26)(27,28)(29,32)(30,31)(33,34,39,37)(35,42,38,40)(36,41,46,45)(43,48,44,47);(3,10)(4,13)(7,9)(8,12)(17,19,23,22)(18,24,26,21)(20,27,29,30)(25,31,32,28)(33,35,39,38)(34,40,37,42)(36,44,46,43)(41,48,45,47);(1,3,6,10)(2,7,5,9)(4,14,13,11)(8,16,12,15)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32)(33,35,39,38)(34,40,37,42)(36,44,46,43)(41,48,45,47);(1,4,6,13)(2,8,5,12)(3,11,10,14)(7,15,9,16)(17,21,23,24)(18,19,26,22)(20,28,29,31)(25,27,32,30)(33,36,39,46)(34,41,37,45)(35,43,38,44)(40,47,42,48);(1,3)(2,9)(4,11)(5,7)(6,10)(8,16)(12,15)(13,14)(17,22,23,19)(18,24,26,21)(20,27,29,30)(25,28,32,31)(33,37,39,34)(35,40,38,42)(36,45,46,41)(43,47,44,48);(1,5,6,2)(3,9,10,7)(4,12,13,8)(11,16,14,15)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32)(33,38,39,35)(34,42,37,40)(36,44,46,43)(41,48,45,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9947713);