#7641494

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4291 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)(3,7,4,6)(9,10,15,19)(11,17,12,16)(13,22,14,18)(20,23,21,24)(25,26,29,31)(27,33,34,30)(28,36,37,32)(35,38,40,39);(1,3,5,4)(2,6,8,7)(9,11,15,12)(10,16,19,17)(13,21,14,20)(18,24,22,23)(25,27,29,34)(26,30,31,33)(28,35,37,40)(32,38,36,39);(1,3,5,4)(2,7,8,6)(9,12,15,11)(10,17,19,16)(13,21,14,20)(18,24,22,23)(26,31)(27,34)(32,36)(35,40);(1,3,5,4)(2,7,8,6)(9,13,15,14)(10,18,19,22)(11,20,12,21)(16,23,17,24)(25,28,29,37)(26,32,31,36)(27,35,34,40)(30,38,33,39);(1,3,5,4)(2,7,8,6)(9,14,15,13)(10,18,19,22)(11,20,12,21)(16,24,17,23);(1,4,5,3)(2,6,8,7)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=g,e^2=gi,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, 7548690);