#9546166

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3640 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9453362);