#8824385

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

3922 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)(18,22)(20,27)(23,28)(26,31);(2,5)(7,9)(8,13)(14,15)(19,25)(23,28)(26,31)(30,32)(33,34,38,36)(35,41,43,39)(37,44,45,40)(42,46,48,47);(3,10)(4,12)(7,9)(8,13)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3,6,10)(2,7,5,9)(4,11,12,16)(8,14,13,15)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(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)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(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=hi,d^2=gi,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, 8731581);