#8898937

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	#46397	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,d^2=gh,e^2=g,f^2=gh,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, 8806133);