#7877878

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4467 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7785074);