#8943869

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	#49489	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4098 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=gh,e^2=g,f^2=g,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, 8851065);