#8587874

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4071 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)(3,8)(4,5)(6,11)(7,9)(10,15)(12,16)(13,14)(17,18,23,26)(19,24,27,21)(20,22,25,31)(28,29,32,30)(34,40)(35,36)(37,45)(47,48);(1,3,7,12)(2,8,9,16)(4,13,10,11)(5,14,15,6)(17,19)(18,24)(20,28)(21,26)(22,29)(23,27)(25,32)(30,31)(33,34,39,40)(35,38,36,43)(37,46,45,41)(42,48,44,47);(2,9)(3,12)(5,15)(11,13)(19,27)(21,24)(28,32)(29,30)(33,35,39,36)(34,38,40,43)(37,44,45,42)(41,48,46,47);(1,4,7,10)(2,5,9,15)(3,11,12,13)(6,16,14,8)(17,20)(18,22)(19,28)(21,30)(23,25)(24,29)(26,31)(27,32)(33,36,39,35)(34,38,40,43)(37,44,45,42)(41,47,46,48);(1,5)(2,10)(3,6)(4,9)(7,15)(8,13)(11,16)(12,14)(17,21)(18,19)(20,30)(22,28)(23,24)(25,29)(26,27)(31,32)(33,37,39,45)(34,41,40,46)(35,42,36,44)(38,47,43,48);(1,6)(2,11)(3,5)(4,8)(7,14)(9,13)(10,16)(12,15)(17,22)(18,25)(19,29)(20,26)(21,28)(23,31)(24,32)(27,30)(33,38)(34,36)(35,40)(37,47)(39,43)(41,44)(42,46)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8495070);