#8402419

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4072 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=i,e^2=g,f^2=gi,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, 8309615);