#8119611

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

4124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8026807);