#6360201

order	512
nilpotency class	3
isoclinism	isoclinic class of #6360176
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#61161	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1839 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:

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

Polycyclic presentation:

2^9|a^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6267397);