#8127420

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

4224 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)(3,6)(4,9)(5,10)(7,11)(8,13)(12,14)(15,16)(18,21)(19,25)(24,29)(26,31)(33,34)(35,40)(36,38)(37,44)(39,42)(41,45)(43,46)(47,48);(2,3)(7,9)(8,10)(14,15)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(35,42)(39,40)(43,48)(46,47);(2,3)(4,11)(8,10)(12,16)(19,25)(20,28)(23,27)(24,29)(33,35,38,42)(34,39,36,40)(37,43,45,48)(41,46,44,47);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19,22,25)(18,23,21,27)(20,26,28,31)(24,30,29,32)(33,35)(34,40)(36,39)(37,43)(38,42)(41,47)(44,46)(45,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20,22,28)(18,24,21,29)(19,26,25,31)(23,30,27,32)(33,36,38,34)(35,39,42,40)(37,44,45,41)(43,46,48,47);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21,22,18)(19,27,25,23)(20,29,28,24)(26,32,31,30)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=g,e^2=gh,f^2=g,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, 8034616);