#7853362

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

4194 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,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, 7760558);