#8720277

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	#44445	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4101 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:

(1,2,7,12)(3,13,15,8)(4,6,11,9)(5,16,14,10)(17,18)(19,21)(20,28)(22,24)(23,30)(25,27)(26,29)(31,32)(34,38)(39,44)(40,45)(46,48);(1,3)(2,8)(4,5)(6,10)(7,15)(9,16)(11,14)(12,13)(17,19,22,27)(18,21,24,25)(20,26,30,32)(23,31,28,29)(35,41)(39,44)(42,47)(46,48);(1,4,7,11)(2,9,12,6)(3,5,15,14)(8,16,13,10)(17,18,22,24)(19,25,27,21)(20,23,30,28)(26,29,32,31)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,4)(2,9)(3,14)(5,15)(6,12)(7,11)(8,10)(13,16)(17,18)(19,25)(20,23)(21,27)(22,24)(26,29)(28,30)(31,32)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,5,7,14)(2,10,12,16)(3,4,15,11)(6,13,9,8)(17,20)(18,23)(19,26)(21,31)(22,30)(24,28)(25,29)(27,32)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,6,7,9)(2,11,12,4)(3,10,15,16)(5,8,14,13)(17,21,22,25)(18,19,24,27)(20,29,30,31)(23,32,28,26)(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=i,c^2=gi,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627473);