#355480

order	512
nilpotency class	3
isoclinism	isoclinic class of #21782
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1558 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)(7,13,9,12)(8,16,18,19)(10,15)(11,14)(17,20);(1,2,3,4)(5,7,11,9)(6,12,14,13)(8,17,18,15)(10,16,20,19);(5,8,10,12,11,18,20,13)(6,7,15,19,14,9,17,16);(5,9,11,7)(6,13,14,12)(8,17,18,15)(10,16,20,19)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=ce,g^b=gi,e^c=ei,f^c=fi,g^c=gi,f^d=fi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262676);