#8479522

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

4166 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)(4,12,14,8)(11,15,13,16)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34,39,37)(35,42,40,36)(38,41,47,45)(43,46,48,44);(1,3,6,5)(2,7,9,10)(4,13,14,11)(8,16,12,15)(18,24)(19,20)(26,29)(27,28)(34,37)(36,42)(41,45)(44,46);(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,46)(47,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,14)(13,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,36,39,42)(34,40,37,35)(38,46,47,44)(41,43,45,48);(1,4,6,14)(2,8,9,12)(3,11,5,13)(7,15,10,16)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,37,39,34)(35,42,40,36)(38,45,47,41)(43,46,48,44);(1,5,6,3)(2,7,9,10)(4,13,14,11)(8,15,12,16)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,38,39,47)(34,41,37,45)(35,43,40,48)(36,44,42,46)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8386718);