#9546019

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=gi,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, 9453215);