#8393447

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	#50677	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4122 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)(3,10)(4,5)(6,7)(8,13)(9,14)(11,12)(15,16)(18,23)(19,26)(25,30)(27,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,48)(42,46)(44,47);(2,7)(4,14)(8,10)(11,15)(17,18,22,23)(19,20,26,24)(21,25,29,30)(27,28,32,31)(33,34)(35,39)(36,43)(37,38)(40,45)(41,48)(42,46)(44,47);(2,7)(5,9)(8,10)(12,16)(21,29)(25,30)(27,32)(28,31)(34,38)(39,46)(40,43)(47,48);(1,3,6,13)(2,8,7,10)(4,15,14,11)(5,12,9,16)(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28)(33,35,37,42)(34,39,38,46)(36,44,45,41)(40,48,43,47);(1,4)(2,9)(3,11)(5,7)(6,14)(8,12)(10,16)(13,15)(17,20,22,24)(18,19,23,26)(21,28,29,31)(25,27,30,32)(33,36,37,45)(34,40,38,43)(35,41,42,44)(39,47,46,48);(1,5,6,9)(2,4,7,14)(3,12,13,16)(8,15,10,11)(17,21,22,29)(18,25,23,30)(19,27,26,32)(20,28,24,31)(33,34)(35,39)(36,40)(37,38)(41,47)(42,46)(43,45)(44,48)

Polycyclic presentation:

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

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, 8300643);