#8747888

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	#36677	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8655084);