#8810168

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3999 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=gh,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8717364);