#8159936

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

4125 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^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^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, 8067132);