#6355311

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355299
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	#60415	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1865 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)(11,15)(12,17)(13,22)(14,18)(16,19)(20,23)(21,24);(1,2,3,5,4,6,7,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,22)(17,24)(18,23);(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,23)(18,24);(9,12,14,13)(10,17,18,22)(11,19,20,21)(15,16,23,24);(10,18)(15,23)(16,24)(17,22)

Polycyclic presentation:

2^9|b^2=f,d^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=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262507);