#8096268

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

4009 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003464);