#8339990

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3842 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)(9,15)(11,16)(13,14)(17,18,23,20)(19,24,22,26)(21,25,30,28)(27,31,29,32)(34,38)(39,44)(40,45)(46,48);(2,7)(3,10)(9,12)(11,13)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(35,41)(39,44)(42,47)(46,48);(2,7)(5,8)(9,12)(14,16)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,3)(2,8)(4,13)(5,7)(6,10)(9,14)(11,15)(12,16)(17,20,23,18)(19,26,22,24)(21,25,30,28)(27,31,29,32)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,4,6,15)(2,9,7,12)(3,11,10,13)(5,14,8,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,5,6,8)(2,3,7,10)(4,14,15,16)(9,11,12,13)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,36,37,43)(34,40,38,45)(35,42,41,47)(39,46,44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=ghi,e^2=g,f^2=hi,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, 8247186);