#8661363

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4143 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8568559);