#8158939

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	#57512	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=ghi,e^2=g,f^2=h,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, 8066135);