#8549575

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

4016 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=g,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

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, 8456771);