#6443355

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437999
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	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

5343 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)(7,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20);(5,6)(7,13,15,10)(8,11)(9,12)(14,19,16,18)(17,20);(1,2,3,4)(5,7)(6,10)(8,16)(9,15)(11,19)(12,13)(14,17)(18,20);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,17)(19,20);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280597, #72283081, #72283331
maximal normal subgroup of	#72280597, #72283081, #72283331

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350551);