#8504785

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	#56880	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3945 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=i,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8411981);