#9546065

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3926 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)(17,18,23,25)(19,28,29,24)(20,30,27,22)(21,26,31,32)(33,34)(35,41)(36,46)(37,43)(38,44)(39,40)(42,47)(45,48);(1,3,7,5)(2,8,10,6)(4,11,13,14)(9,15,16,12)(17,19)(18,24)(20,21)(22,26)(23,29)(25,28)(27,31)(30,32)(33,35,40,37)(34,41,39,43)(36,38,47,45)(42,48,46,44);(3,5)(6,8)(11,14)(12,15)(18,25)(19,29)(20,27)(26,32)(33,36,40,47)(34,42,39,46)(35,45,37,38)(41,44,43,48);(1,4,7,13)(2,9,10,16)(3,11,5,14)(6,12,8,15)(17,20,23,27)(18,22,25,30)(19,21,29,31)(24,26,28,32)(33,37,40,35)(34,43,39,41)(36,45,47,38)(42,44,46,48);(1,5,7,3)(2,6,10,8)(4,11,13,14)(9,15,16,12)(17,21)(18,26)(19,27)(20,29)(22,28)(23,31)(24,30)(25,32)(33,38,40,45)(34,44,39,48)(35,36,37,47)(41,46,43,42);(1,6)(2,3)(4,12)(5,10)(7,8)(9,11)(13,15)(14,16)(17,22)(18,27)(19,26)(20,25)(21,28)(23,30)(24,31)(29,32)(33,39,40,34)(35,43,37,41)(36,42,47,46)(38,48,45,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=i,e^2=gi,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, 9453261);