#8691837

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	#57996	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4217 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)(4,10)(5,14)(6,12)(7,13)(9,15)(11,16)(17,18,22,23)(19,24,20,25)(21,26,30,29)(27,31,28,32)(33,34,37,40)(35,38,42,45)(36,39,43,46)(41,44,47,48);(1,3,7,6)(2,8,13,12)(4,11,15,14)(5,10,16,9)(18,23)(24,25)(26,29)(31,32)(33,35,37,42)(34,38,40,45)(36,41,43,47)(39,44,46,48);(1,4)(2,9)(3,14)(5,12)(6,11)(7,15)(8,16)(10,13)(17,19,22,20)(18,24,23,25)(21,27,30,28)(26,31,29,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,4)(2,10)(3,11)(5,12)(6,14)(7,15)(8,16)(9,13)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,32)(28,31)(33,35)(34,38)(36,41)(37,42)(39,44)(40,45)(43,47)(46,48);(1,5)(2,11)(3,10)(4,12)(6,9)(7,16)(8,15)(13,14)(17,20,22,19)(18,25,23,24)(21,27,30,28)(26,31,29,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,6,7,3)(2,12,13,8)(4,11,15,14)(5,10,16,9)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,36,37,43)(34,39,40,46)(35,41,42,47)(38,44,45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,e^2=g,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, 8599033);