#8957560

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3833 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:

(1,2,7,4)(3,8,10,6)(5,9,13,12)(11,15,16,14)(17,18)(19,25)(20,21)(22,30)(23,24)(26,31)(27,29)(28,32)(35,38)(40,42)(43,45)(47,48);(1,3,7,10)(2,8,4,6)(5,11,13,16)(9,15,12,14)(18,24)(21,25)(26,30)(29,32)(33,34,39,36)(35,40,38,42)(37,44,46,41)(43,48,45,47);(3,10)(5,13)(6,8)(9,12)(17,19,23,20)(18,21,24,25)(22,28,31,27)(26,32,30,29)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,4,7,2)(3,6,10,8)(5,9,13,12)(11,15,16,14)(17,20,23,19)(18,25,24,21)(22,28,31,27)(26,32,30,29)(33,36,39,34)(35,42,38,40)(37,44,46,41)(43,48,45,47);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,21,23,25)(18,20,24,19)(22,29,31,32)(26,28,30,27)(33,37,39,46)(34,41,36,44)(35,43,38,45)(40,47,42,48);(1,6)(2,3)(4,10)(5,14)(7,8)(9,11)(12,16)(13,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(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=ghi,c^2=h,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8864756);