#8963167

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	#42409	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=i,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8870363);