#114372

order	512
nilpotency class	3
isoclinism	isoclinic class of #100027
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×AGL(1,ℤ/8)	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,f^b=fi,g^b=ghi,f^c=fi

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, 21568);