#8898942

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	#47004	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4178 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)(3,12)(4,15)(5,6)(7,9)(8,14)(10,13)(11,16)(17,18)(19,24)(20,25)(21,30)(22,23)(26,27)(28,31)(29,32)(33,34,39,38)(35,41,36,40)(37,42,46,45)(43,48,44,47);(1,3,7,14)(2,8,9,12)(4,11,13,5)(6,10,16,15)(17,18)(19,24)(20,21)(22,23)(25,30)(26,32)(27,29)(28,31)(33,35,39,36)(34,40,38,41)(37,43,46,44)(42,47,45,48);(2,9)(6,16)(8,12)(10,15)(18,23)(21,25)(24,31)(27,32)(33,35,39,36)(34,41,38,40)(37,43,46,44)(42,48,45,47);(1,4)(2,10)(3,11)(5,14)(6,12)(7,13)(8,16)(9,15)(17,19,22,28)(18,24,23,31)(20,29,30,26)(21,27,25,32)(33,36,39,35)(34,41,38,40)(37,43,46,44)(42,47,45,48);(1,5,7,11)(2,6,9,16)(3,13,14,4)(8,15,12,10)(17,20)(18,25)(19,26)(21,23)(22,30)(24,27)(28,29)(31,32)(33,37,39,46)(34,42,38,45)(35,43,36,44)(40,47,41,48);(1,6)(2,11)(3,10)(4,12)(5,9)(7,16)(8,13)(14,15)(17,21,22,25)(18,20,23,30)(19,27,28,32)(24,29,31,26)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8806138);