#354923

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1762 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=ehi,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,c^b=ci,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262119);