#8838372

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	#49899	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4055 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)(3,7)(4,8)(5,14)(6,10)(9,15)(11,12)(13,16)(17,18)(19,20)(21,29)(22,26)(23,24)(25,30)(27,31)(28,32)(33,34,39,36)(35,42,40,37)(38,44,46,41)(43,45,47,48);(1,3)(2,7)(4,5)(6,11)(8,14)(9,16)(10,12)(13,15)(17,18,22,26)(19,23,24,20)(21,29,30,25)(27,28,32,31)(35,40)(37,42)(38,46)(41,44);(3,11)(4,13)(7,12)(8,16)(17,19,22,24)(18,23,26,20)(21,27,30,32)(25,31,29,28)(33,35,39,40)(34,37,36,42)(38,43,46,47)(41,45,44,48);(1,4,6,13)(2,8,10,16)(3,5,11,15)(7,14,12,9)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,29)(30,32)(33,36,39,34)(35,37,40,42)(38,44,46,41)(43,45,47,48);(1,5)(2,9)(3,4)(6,15)(7,16)(8,12)(10,14)(11,13)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,32,29,27)(33,37)(34,40)(35,36)(38,45)(39,42)(41,47)(43,44)(46,48);(1,2)(3,12)(4,8)(5,9)(6,10)(7,11)(13,16)(14,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,38,39,46)(34,41,36,44)(35,43,40,47)(37,45,42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=i,e^2=g,f^2=gh,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, 8745568);