#6443331

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437963
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

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

5226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 20 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215857, #72219690, #72227253, …
maximal normal subgroup of	#72215857, #72219690, #72227253, …
maximal quotient of	#72264510, #72280609, C₂×#6443331

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350527);