#8267562

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3908 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)(3,11)(4,13)(5,10)(6,8)(7,14)(9,15)(12,16)(17,18,22,21)(19,23,27,25)(20,28,29,24)(26,31,32,30);(1,3,6,14)(2,7,8,11)(4,10,15,12)(5,13,16,9)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34)(35,38)(36,40)(37,45)(39,42)(41,47)(43,48)(44,46);(2,8)(5,16)(7,11)(9,13)(18,21)(23,25)(24,28)(30,31)(33,35,39,36)(34,40,42,38)(37,43,47,44)(41,48,45,46);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,16)(10,15)(17,19,22,27)(18,23,21,25)(20,26,29,32)(24,30,28,31)(33,36,39,35)(34,40,42,38)(37,44,47,43)(41,48,45,46);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,7,16,11)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,37,39,47)(34,41,42,45)(35,43,36,44)(38,46,40,48);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,7,15,11)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)(33,38,39,40)(34,35,42,36)(37,46,47,48)(41,43,45,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=ghi,d^2=i,e^2=gi,f^2=gi,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, 8174758);