#8359681

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

4251 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,16)(14,15)(17,18,23,20)(19,24,26,22)(21,25,31,28)(27,32,29,30)(33,34,39,41)(35,42,36,37)(38,40,46,48)(43,47,44,45);(2,7)(3,10)(4,13)(15,16)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,30,28,32)(35,36)(37,42)(43,44)(45,47);(2,7)(5,8)(9,12)(15,16)(19,26)(22,24)(27,29)(30,32)(33,35,39,36)(34,37,41,42)(38,44,46,43)(40,47,48,45);(1,3)(2,8)(4,14)(5,7)(6,10)(9,15)(11,13)(12,16)(17,20,23,18)(19,22,26,24)(21,25,31,28)(27,32,29,30)(33,36,39,35)(34,37,41,42)(38,44,46,43)(40,45,48,47);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,37)(34,36)(35,41)(38,45)(39,42)(40,44)(43,48)(46,47);(1,5,6,8)(2,3,7,10)(4,15,13,16)(9,11,12,14)(17,22)(18,19)(20,26)(21,30)(23,24)(25,27)(28,29)(31,32)(33,38)(34,40)(35,43)(36,44)(37,45)(39,46)(41,48)(42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=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, 8266877);