#8824348

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	#49994	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4133 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)(3,8)(4,10)(5,6)(7,9)(11,15)(12,16)(13,14)(18,23)(19,24)(25,29)(26,31)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(1,3)(2,8)(4,11)(5,7)(6,9)(10,15)(12,13)(14,16)(17,18,22,23)(19,20,24,27)(21,29,30,25)(26,32,31,28)(35,41)(37,45)(39,42)(40,44);(2,9)(3,5)(4,12)(14,15)(19,24)(20,27)(26,31)(28,32)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,4,7,12)(2,10,9,16)(3,11,5,13)(6,14,8,15)(17,19,22,24)(18,20,23,27)(21,26,30,31)(25,28,29,32)(33,34,38,36)(35,42,41,39)(37,40,45,44)(43,48,47,46);(1,5,7,3)(2,6,9,8)(4,13,12,11)(10,14,16,15)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32)(33,36,38,34)(35,39,41,42)(37,44,45,40)(43,46,47,48);(1,6)(2,3)(4,14)(5,9)(7,8)(10,11)(12,15)(13,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=gi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8731544);