#6347327

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347305
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	C₂²×C₄	C₂⁵
prime radical	#62788	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2988 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271341, #72271558, #72271575
maximal normal subgroup of	#72271341, #72271558, #72271575

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254523);