#8119377

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

4218 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34,38,41)(35,42,36,39)(37,45,46,40)(43,47,44,48);(2,7)(3,9)(10,13)(11,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(35,36)(39,42)(43,44)(47,48);(2,7)(4,8)(10,13)(12,15)(20,28)(23,30)(26,32)(29,31)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,15,13,12)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(1,5,6,14)(2,10,7,13)(3,11,9,16)(4,12,8,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,37,38,46)(34,40,41,45)(35,43,36,44)(39,47,42,48)

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8026573);