#8127441

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

4116 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8034637);