#466610

order	512
nilpotency class	3
isoclinism	isoclinic class of #456400
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

979 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

(5,6)(7,10)(8,9)(11,12)(13,14,18,21)(15,22,20,17)(16,24,26,19)(23,25,27,28);(1,2,3,4)(6,9)(10,12);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,14,18,20,22,21)(16,23,25,19,26,27,28,24);(7,11)(10,12)(13,16,17,25,18,26,22,28)(14,19,20,27,21,24,15,23)

Polycyclic presentation:

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

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, 373806);