#8159134

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3868 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066330);