#8159407

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

4199 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=gi,e^2=g,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, 8066603);