#8826367

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	#35053	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4131 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=gh,e^2=h,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, 8733563);