#8720681

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	#56386	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4228 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)(3,11)(4,8)(5,7)(6,10)(9,14)(12,16)(13,15)(17,18)(19,22)(20,30)(21,32)(23,27)(24,29)(25,31)(26,28)(33,34,39,42)(35,38,40,36)(37,41,47,48)(43,46,45,44);(1,3,6,5)(2,7,10,11)(4,12,14,13)(8,15,9,16)(17,19)(18,24)(20,21)(22,27)(23,29)(25,32)(26,30)(28,31)(33,35)(34,38)(36,42)(37,45)(39,40)(41,44)(43,47)(46,48);(1,4,6,14)(2,8,10,9)(3,12,5,13)(7,15,11,16)(17,20,23,31)(18,25,27,30)(19,21,29,28)(22,26,24,32)(33,36,39,38)(34,40,42,35)(37,46,47,44)(41,43,48,45);(1,4)(2,9)(3,12)(5,13)(6,14)(7,16)(8,10)(11,15)(17,20)(18,25)(19,28)(21,29)(22,32)(23,31)(24,26)(27,30)(33,37,39,47)(34,41,42,48)(35,43,40,45)(36,44,38,46);(1,5,6,3)(2,7,10,11)(4,13,14,12)(8,15,9,16)(17,21,23,28)(18,26,27,32)(19,20,29,31)(22,25,24,30)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,47)(46,48);(1,2)(3,7)(4,8)(5,11)(6,10)(9,14)(12,15)(13,16)(17,22,23,24)(18,19,27,29)(20,26,31,32)(21,30,28,25)(33,38,39,36)(34,40,42,35)(37,44,47,46)(41,43,48,45)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gi,d^2=i,e^2=g,f^2=h,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, 8627877);