#9206532

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3872 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9113728);