#8696750

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	#36369	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8603946);