#9668269

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	#2828	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3803 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)(3,9)(4,12)(5,6)(7,10)(8,14)(11,15)(13,16)(17,18)(19,25)(20,22)(21,24)(23,27)(26,31)(28,29)(30,32)(33,34,39,36)(35,40,45,43)(37,41,44,46)(38,42,48,47);(2,6)(7,9)(8,12)(15,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,35,39,45)(34,40,36,43)(37,38,44,48)(41,42,46,47);(2,6)(3,10)(8,12)(11,13)(21,29)(24,28)(26,32)(30,31)(33,34,39,36)(35,43,45,40)(37,46,44,41)(38,42,48,47);(1,3)(2,7)(4,13)(5,10)(6,9)(8,16)(11,14)(12,15)(17,20,22,18)(19,25,27,23)(21,24,29,28)(26,30,32,31)(33,36,39,34)(35,43,45,40)(37,41,44,46)(38,42,48,47);(1,2)(3,9)(4,8)(5,6)(7,10)(11,16)(12,14)(13,15)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,37,39,44)(34,41,36,46)(35,38,45,48)(40,42,43,47);(1,4,5,14)(2,8,6,12)(3,11,10,13)(7,15,9,16)(17,19)(18,25)(20,23)(21,26)(22,27)(24,31)(28,30)(29,32)(33,38)(34,42)(35,44)(36,47)(37,45)(39,48)(40,46)(41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575465);