#8374218

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3829 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:

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

Polycyclic presentation:

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

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, 8281414);