#9007339

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	#46407	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3914 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)(4,9,12,16)(11,15,14,13)(17,18,23,28)(19,29,30,24)(20,25,27,22)(21,26,31,32)(34,38)(35,43)(36,46)(42,48);(1,3,7,5)(2,8,10,6)(4,11,12,14)(9,15,16,13)(19,30)(21,31)(24,29)(26,32)(33,34,39,38)(35,44,43,40)(36,41,46,47)(37,48,45,42);(3,5)(4,12)(6,8)(9,16)(17,19,23,30)(18,24,28,29)(20,21,27,31)(22,26,25,32)(33,35,39,43)(34,40,38,44)(36,45,46,37)(41,48,47,42);(1,4)(2,9)(3,11)(5,14)(6,13)(7,12)(8,15)(10,16)(17,20,23,27)(18,25,28,22)(19,31,30,21)(24,26,29,32)(33,36)(34,41)(35,45)(37,43)(38,47)(39,46)(40,48)(42,44);(1,5,7,3)(2,6,10,8)(4,11,12,14)(9,15,16,13)(17,21,23,31)(18,26,28,32)(19,20,30,27)(22,29,25,24)(33,37,39,45)(34,42,38,48)(35,36,43,46)(40,41,44,47);(1,6)(2,3)(4,13)(5,10)(7,8)(9,11)(12,15)(14,16)(17,22,23,25)(18,27,28,20)(19,32,30,26)(21,29,31,24)(33,38,39,34)(35,44,43,40)(36,47,46,41)(37,48,45,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=gh,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=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8914535);