#8700381

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,d^2=gh,e^2=h,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, 8607577);