#449879

order	512
nilpotency class	3
isoclinism	isoclinic class of #427387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1263 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 20 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357075);