#7640273

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4485 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)(3,7)(4,8)(5,6)(9,10)(11,18)(12,15)(13,21)(14,17)(16,20)(19,23)(22,24)(25,26,29,32)(27,30,34,38)(28,31,36,40)(33,37,35,39);(2,6)(7,8)(9,11)(10,16)(12,18)(13,22)(14,19)(15,20)(17,24)(21,23)(25,27)(26,30)(28,35)(29,34)(31,39)(32,38)(33,36)(37,40);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,18,20,16)(13,21,14,17)(19,24,22,23)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(1,3,5,4)(2,7,6,8)(9,13,15,14)(10,17,12,21)(11,19,20,22)(16,23,18,24)(27,34)(30,38)(33,35)(37,39);(1,3,5,4)(2,7,6,8)(9,14,15,13)(10,17,12,21)(11,19,20,22)(16,24,18,23)(28,36)(31,40)(33,35)(37,39);(1,4,5,3)(2,8,6,7)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7547469);