#8119557

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	#39300	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3929 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,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, 8026753);