#8750943

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	#53073	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=i,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, 8658139);