#8772971

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

4190 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,d^2=gh,e^2=h,f^2=gh,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, 8680167);