#8187019

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	#38859	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(17,18)(19,21)(20,28)(22,24)(23,27)(25,30)(26,31)(29,32)(33,34,37,38)(35,41,43,39)(36,40,44,45)(42,47,48,46);(2,7)(3,10)(9,14)(11,16)(17,19)(18,23)(20,26)(21,24)(22,27)(25,31)(28,29)(30,32)(34,38)(39,41)(40,45)(46,47);(5,13)(9,14)(11,16)(12,15)(18,24)(21,23)(25,28)(29,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,14)(13,16)(17,20,22,30)(18,25,24,28)(19,26,27,32)(21,29,23,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4,6,8)(2,3,7,10)(5,12,13,15)(9,11,14,16)(17,19)(18,23)(20,26)(21,24)(22,27)(25,31)(28,29)(30,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,6,13)(2,9,7,14)(3,11,10,16)(4,12,8,15)(17,21,22,23)(18,19,24,27)(20,29,30,31)(25,26,28,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=i,d^2=hi,e^2=gh,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, 8094215);