#9506888

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	#55620	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3851 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=gi,f^2=g,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, 9414084);