#8407119

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	#52586	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|b^2=i,d^2=g,e^2=gh,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, 8314315);