#9546049

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

3646 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)(19,25)(20,30)(21,27)(22,28)(23,24)(26,31)(29,32)(33,34)(35,37)(36,44)(38,42)(39,40)(41,47)(43,46)(45,48);(1,3,7,5)(2,8,10,6)(4,11,13,14)(9,15,16,12)(17,19,24,21)(18,25,23,27)(20,22,31,29)(26,32,30,28)(34,40)(35,42)(41,44)(43,48);(3,5)(6,8)(11,14)(12,15)(17,20,24,31)(18,26,23,30)(19,29,21,22)(25,28,27,32)(33,35,39,42)(34,37,40,38)(36,43,47,48)(41,45,44,46);(1,4,7,13)(2,9,10,16)(3,11,5,14)(6,12,8,15)(17,21,24,19)(18,27,23,25)(20,29,31,22)(26,28,30,32)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,5,7,3)(2,6,10,8)(4,11,13,14)(9,15,16,12)(17,22,24,29)(18,28,23,32)(19,20,21,31)(25,30,27,26)(33,37,39,38)(34,35,40,42)(36,45,47,46)(41,43,44,48);(1,6)(2,3)(4,12)(5,10)(7,8)(9,11)(13,15)(14,16)(17,23,24,18)(19,27,21,25)(20,26,31,30)(22,32,29,28)(33,38,39,37)(34,42,40,35)(36,46,47,45)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=i,e^2=i,f^2=gh,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, 9453245);