#8127549

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

4078 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,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,d^c=dgi,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, 8034745);