#106409

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#379
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

974 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=ci,d^b=dh,e^c=eh,f^c=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13605);