#9546142

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	#57091	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3842 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)(18,23)(20,28)(24,32)(25,29)(33,34)(35,43)(36,41)(37,42)(38,39)(40,44)(45,48)(46,47);(1,3,7,5)(2,8,10,6)(4,11,13,14)(9,15,16,12)(17,18)(19,24)(20,21)(22,23)(25,30)(26,29)(27,32)(28,31)(35,44)(37,45)(40,43)(42,48);(3,5)(6,8)(11,14)(12,15)(17,19,22,27)(18,24,23,32)(20,25,28,29)(21,30,31,26)(33,35,39,44)(34,40,38,43)(36,37,47,45)(41,48,46,42);(1,4,7,13)(2,9,10,16)(3,11,5,14)(6,12,8,15)(17,20)(18,21)(19,25)(22,28)(23,31)(24,30)(26,32)(27,29)(33,36,39,47)(34,41,38,46)(35,45,44,37)(40,42,43,48);(1,5,7,3)(2,6,10,8)(4,11,13,14)(9,15,16,12)(17,19)(18,24)(20,29)(21,26)(22,27)(23,32)(25,28)(30,31)(33,37,39,45)(34,42,38,48)(35,36,44,47)(40,46,43,41);(1,6)(2,3)(4,12)(5,10)(7,8)(9,11)(13,15)(14,16)(17,21)(18,20)(19,26)(22,31)(23,28)(24,29)(25,32)(27,30)(33,38,39,34)(35,40,44,43)(36,46,47,41)(37,48,45,42)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=i,e^2=i,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, 9453338);