#109795

order	512
nilpotency class	3
isoclinism	isoclinic class of #9159
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

708 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=ghi,d^b=di,e^c=ehi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16991);