#8824361

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

4047 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=gi,e^2=ghi,f^2=hi,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, 8731557);