#8873607

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3884 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)(3,9)(4,13)(5,14)(6,8)(7,12)(10,16)(11,15)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30)(35,38)(40,42)(43,45)(47,48);(1,3,6,12)(2,7,8,9)(4,10,14,15)(5,16,13,11)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(2,8)(4,14)(7,9)(10,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,4)(2,5)(3,10)(6,14)(7,16)(8,13)(9,11)(12,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,3)(2,7)(4,15)(5,11)(6,12)(8,9)(10,14)(13,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,5)(2,4)(3,11)(6,13)(7,15)(8,14)(9,10)(12,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,38,39,35)(34,42,36,40)(37,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=ghi,e^2=ghi,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, 8780803);