#7853316

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4192 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=h,e^2=gh,f^2=g,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, 7760512);