#6442733

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437632
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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

4899 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)(13,18)(14,17)(15,19)(16,20);(8,15)(10,17)(11,19)(13,16);(1,2,3,4)(5,7)(6,10)(8,16)(9,14)(11,20)(12,17)(13,15)(18,19);(5,7)(6,10)(8,13)(9,14)(11,18)(12,17)(15,16)(19,20);(5,8,9,15)(6,11,12,19)(7,13,14,16)(10,18,17,20)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349929);