#441074

order	512
nilpotency class	3
isoclinism	isoclinic class of #417460
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×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1651 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)(13,14,19,22,17,16,18,21)(15,23,26,27,24,20,25,28);(2,3)(5,6,9,10)(7,11,8,12)(13,15,17,24)(14,20,16,23)(18,25,19,26)(21,27,22,28);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16,17,14)(15,20,24,23)(18,21,19,22)(25,28,26,27);(1,3,4,2)(5,8,9,7)(6,11,10,12)(13,14,17,16)(15,23,24,20)(18,21,19,22)(25,28,26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348270);