#8393448

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	#50801	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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)(3,11,13,8)(4,12,14,9)(5,10,16,15)(17,18,23,21)(19,26,24,20)(22,25,30,29)(27,32,31,28)(33,34)(35,39)(36,43)(37,38)(40,45)(41,48)(42,46)(44,47);(2,7)(4,14)(8,11)(10,15)(17,19,23,24)(18,20,21,26)(22,27,30,31)(25,28,29,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,48)(42,46)(44,47);(2,7)(5,16)(8,11)(9,12)(22,30)(25,29)(27,31)(28,32)(34,38)(39,46)(40,43)(47,48);(1,3,6,13)(2,8,7,11)(4,15,14,10)(5,12,16,9)(17,20,23,26)(18,24,21,19)(22,28,30,32)(25,31,29,27)(33,35,37,42)(34,39,38,46)(36,44,45,41)(40,48,43,47);(1,4)(2,9)(3,10)(5,11)(6,14)(7,12)(8,16)(13,15)(17,21,23,18)(19,20,24,26)(22,29,30,25)(27,28,31,32)(33,36,37,45)(34,40,38,43)(35,41,42,44)(39,47,46,48);(1,5,6,16)(2,10,7,15)(3,12,13,9)(4,11,14,8)(17,22,23,30)(18,25,21,29)(19,27,24,31)(20,28,26,32)(33,34)(35,39)(36,40)(37,38)(41,47)(42,46)(43,45)(44,48)

Polycyclic presentation:

2^9|a^2=g,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, 8300644);