#8901486

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	#49114	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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)(3,11)(4,9)(5,12)(6,7)(8,14)(10,16)(13,15)(33,34,39,38)(35,41,45,40)(36,46,44,42)(37,43,48,47);(2,7)(4,15)(8,11)(10,12)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,32)(28,30)(33,35,39,45)(34,40,38,41)(36,37,44,48)(42,43,46,47);(2,7)(5,16)(8,11)(9,13)(18,22)(23,29)(24,30)(31,32)(33,35,39,45)(34,41,38,40)(36,37,44,48)(42,47,46,43);(1,3,6,14)(2,8,7,11)(4,10,15,12)(5,13,16,9)(17,19,21,26)(18,23,22,29)(20,27,28,25)(24,32,30,31)(33,36)(34,42)(35,37)(38,46)(39,44)(40,43)(41,47)(45,48);(1,4,6,15)(2,9,7,13)(3,12,14,10)(5,8,16,11)(17,20,21,28)(18,24,22,30)(19,25,26,27)(23,31,29,32)(33,37)(34,43)(35,44)(36,45)(38,47)(39,48)(40,46)(41,42);(1,5,6,16)(2,10,7,12)(3,13,14,9)(4,8,15,11)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,32)(28,30)(33,38,39,34)(35,41,45,40)(36,46,44,42)(37,47,48,43)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=hi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=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, 8808682);