#8210208

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	#38836	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3938 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:

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

Polycyclic presentation:

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

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, 8117404);