#8901461

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	#58965	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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:

(17,18,22,26)(19,27,28,23)(20,24,29,32)(21,25,30,31)(34,39)(35,40)(37,45)(43,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(19,28)(21,30)(23,27)(25,31)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,47)(46,48);(2,6)(7,13)(8,14)(15,16)(17,19,22,28)(18,23,26,27)(20,21,29,30)(24,31,32,25)(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)(17,20)(18,24)(19,21)(22,29)(23,31)(25,27)(26,32)(28,30)(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)(17,21)(18,25)(19,29)(20,28)(22,30)(23,24)(26,31)(27,32)(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)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,32)(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=h,d^2=ghi,e^2=ghi,f^2=g,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, 8808657);