#8193918

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	#52415	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4079 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=hi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8101114);