#8701677

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

3809 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(2,7)(3,10)(9,16)(11,13)(17,18,23,22)(19,27,29,24)(20,30,28,25)(21,26,31,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(17,19,23,29)(18,24,22,27)(20,21,28,31)(25,26,30,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,3)(2,8)(4,7)(5,13)(6,10)(9,12)(11,15)(14,16)(17,20,23,28)(18,25,22,30)(19,21,29,31)(24,26,27,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,13,16,11)(17,21)(18,26)(19,28)(20,29)(22,32)(23,31)(24,30)(25,27)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(1,5,6,15)(2,9,7,16)(3,11,10,13)(4,12,8,14)(17,22,23,18)(19,27,29,24)(20,25,28,30)(21,26,31,32)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8608873);