#8393431

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

4156 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)(3,11)(4,13)(5,10)(6,7)(8,14)(9,15)(12,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,48)(42,46)(44,47);(2,7)(4,15)(8,11)(10,12)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34)(35,39)(36,43)(37,38)(40,45)(41,48)(42,46)(44,47);(2,7)(5,16)(8,11)(9,13)(20,28)(23,30)(26,32)(29,31)(34,38)(39,46)(40,43)(47,48);(1,3,6,14)(2,8,7,11)(4,10,15,12)(5,13,16,9)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,35,37,42)(34,39,38,46)(36,44,45,41)(40,48,43,47);(1,4,6,15)(2,9,7,13)(3,12,14,10)(5,8,16,11)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,36,37,45)(34,40,38,43)(35,41,42,44)(39,47,46,48);(1,5,6,16)(2,10,7,12)(3,13,14,9)(4,8,15,11)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(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=h,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, 8300627);