#8726196

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3966 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8633392);