#9232796

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=ghi,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,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, 9139992);