#8393678

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

4123 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300874);