#6435591

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433075
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60415	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(3,7)(6,8)(9,10,14,18,15,20,13,19)(11,21,22,16,23,24,17,12);(1,2)(3,6)(4,5)(7,8)(9,11,14,22,15,23,13,17)(10,16,19,21,20,12,18,24);(2,5)(6,8)(9,12)(10,17)(11,19)(13,21)(14,24)(15,16)(18,23)(20,22);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,18,20,19)(11,17,23,22)(12,21,16,24);(9,13,15,14)(10,19,20,18)(11,22,23,17)(12,21,16,24)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6342787);