#6438157

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,g^c=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345353);