#8901583

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	#46804	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=ghi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8808779);