#8121422

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	#40285	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=gi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8028618);