#6355399

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355367
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60561	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1834 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)(9,10,14,16)(11,13,18,23)(12,19,22,17)(15,20,24,21);(1,2,3,5,4,6,7,8)(9,11,13,16,14,18,23,10)(12,17,20,24,22,19,21,15);(9,12)(10,15)(11,17)(13,20)(14,22)(16,24)(18,19)(21,23);(2,6)(5,8);(9,13,14,23)(10,11,16,18)(12,20,22,21)(15,17,24,19)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262595);