#441050

order	512
nilpotency class	3
isoclinism	isoclinic class of #417575
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1563 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348246);