#8806818

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3777 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714014);