#8720580

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4110 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=i,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8627776);