#10188291

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3929 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:

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

Polycyclic presentation:

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

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, 10095487);