#8662935

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4100 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8570131);