#8533083

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3901 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8440279);