#8824392

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

3910 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=hi,e^2=i,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, 8731588);