#8118641

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

4195 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=ghi,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, 8025837);