#8678986

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3801 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=h,e^2=g,f^2=gi,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, 8586182);