#8202260

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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)(4,12)(5,6)(7,11)(8,14)(10,15)(13,16)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(34,36)(39,41)(40,44)(46,47);(1,3)(2,7)(4,13)(5,9)(6,11)(8,16)(10,14)(12,15)(19,27)(20,30)(23,32)(24,29)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(1,2,6,5)(3,7,11,9)(4,8,14,12)(10,15,13,16)(17,19)(18,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,4)(2,8)(3,10)(5,12)(6,14)(7,15)(9,16)(11,13)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,5,6,2)(3,7,11,9)(4,12,14,8)(10,15,13,16)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47);(1,3)(2,9)(4,10)(5,7)(6,11)(8,16)(12,15)(13,14)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)

Polycyclic presentation:

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

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, 8109456);