#8749724

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3879 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)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,31)(29,30)(34,38)(35,41)(36,44)(46,48);(2,6)(7,9)(8,12)(15,16)(17,19,22,21)(18,23,25,26)(20,27,28,29)(24,30,31,32)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(2,6)(7,9)(8,12)(15,16)(20,28)(24,31)(27,29)(30,32)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,2)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,20,22,28)(18,24,25,31)(19,27,21,29)(23,30,26,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,21,22,19)(18,23,25,26)(20,27,28,29)(24,32,31,30)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=gh,e^2=h,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, 8656920);