#8418151

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3917 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18,23,27)(19,26,29,22)(20,24,21,25)(28,30,31,32)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(1,3)(2,8)(4,6)(5,15)(7,12)(9,13)(10,14)(11,16)(17,19,23,29)(18,22,27,26)(20,28,21,31)(24,32,25,30)(34,38)(35,42)(40,44)(43,48);(3,12)(6,13)(8,16)(10,15)(17,20,23,21)(18,24,27,25)(19,28,29,31)(22,32,26,30)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,4)(2,5)(3,13)(6,12)(7,9)(8,10)(11,14)(15,16)(17,21,23,20)(18,25,27,24)(19,28,29,31)(22,32,26,30)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,5)(2,9)(3,10)(4,11)(6,8)(7,14)(12,15)(13,16)(17,19)(18,26)(20,28)(21,31)(22,27)(23,29)(24,30)(25,32)(33,36,37,45)(34,40,38,44)(35,43,42,48)(39,46,41,47);(1,6,7,13)(2,10,11,15)(3,4,12,9)(5,8,14,16)(17,22,23,26)(18,19,27,29)(20,30,21,32)(24,31,25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8325347);