#9027323

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	#52387	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=h,e^2=h,f^2=i,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, 8934519);