#8873781

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

3939 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780977);