#6516742

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499992
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62833	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3360 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268177, #72268178, #72268187
maximal normal subgroup of	#72268177, #72268178, #72268187

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6423938);