#8824414

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3926 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=hi,e^2=ghi,f^2=gi,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, 8731610);