#8180409

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4078 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=g,e^2=gh,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, 8087605);