#8291664

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	#50059	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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)(3,5)(4,12)(6,7)(8,10)(9,15)(11,16)(13,14)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(34,36)(40,43)(41,42)(47,48);(2,7)(3,10)(9,12)(11,13)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,34,39,36)(35,43,46,40)(37,41,38,42)(44,48,45,47);(2,7)(5,8)(9,12)(14,16)(19,25)(23,26)(27,31)(30,32)(33,35,39,46)(34,40,36,43)(37,44,38,45)(41,47,42,48);(1,3)(2,8)(4,13)(5,7)(6,10)(9,14)(11,15)(12,16)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,36,39,34)(35,43,46,40)(37,42,38,41)(44,48,45,47);(1,4,6,15)(2,9,7,12)(3,11,10,13)(5,14,8,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,39,38)(34,41,36,42)(35,44,46,45)(40,47,43,48);(1,5,6,8)(2,3,7,10)(4,14,15,16)(9,11,12,13)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,38,39,37)(34,42,36,41)(35,45,46,44)(40,48,43,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8198860);