#9166432

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	#2657	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4021 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9073628);