#8291672

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

4074 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)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,34,39,38)(35,41,36,40)(37,42,46,45)(43,48,44,47);(2,7)(3,12)(9,14)(10,13)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35,39,36)(34,40,38,41)(37,43,46,44)(42,47,45,48);(2,7)(5,16)(8,11)(9,14)(19,25)(23,26)(27,31)(30,32)(33,35,39,36)(34,41,38,40)(37,43,46,44)(42,48,45,47);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,36,39,35)(34,41,38,40)(37,43,46,44)(42,47,45,48);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,37,39,46)(34,42,38,45)(35,43,36,44)(40,47,41,48);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8198868);