#8826223

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3955 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8733419);