#10366162

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	#53742	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4163 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

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, 10273358);