#8889447

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	#45816	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 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)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(18,20)(21,27)(25,28)(29,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,2)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,18,23,20)(19,21,24,27)(22,28,30,25)(26,32,31,29)(36,44)(39,46)(42,48)(45,47);(2,6)(7,9)(8,12)(14,15)(17,19,23,24)(18,21,20,27)(22,26,30,31)(25,29,28,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20,23,18)(19,21,24,27)(22,28,30,25)(26,29,31,32);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21)(18,24)(19,20)(22,29)(23,27)(25,31)(26,28)(30,32)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,22,23,30)(18,25,20,28)(19,26,24,31)(21,29,27,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8796643);