#8271330

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

4069 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:

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

Polycyclic presentation:

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

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, 8178526);