#6353663

order	512
nilpotency class	3
isoclinism	isoclinic class of #19162
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

1868 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#135×AGL(1,ℤ/8), #72342644, #72342647
maximal normal subgroup of	#135×AGL(1,ℤ/8), #72342644, #72342647

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260859);