#8179625

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

4146 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8086821);