#6349545

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3863 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)(3,8)(4,6)(5,7)(10,15)(11,19)(12,13)(17,18)(20,21)(22,23);(2,6)(3,4)(7,8)(9,10,14,18)(11,16,20,24)(12,15,13,17)(19,22,21,23);(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,23)(18,24);(2,7)(6,8);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,17,18,15)(11,19,20,21)(16,23,24,22)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257890, #72257895, #72257915
maximal normal subgroup of	#72257890, #72257895, #72257915
maximal quotient of	#72264347, #72264352

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256741);