#109386

order	512
nilpotency class	3
isoclinism	isoclinic class of #8674
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

642 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ei,d^2=i,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,f^b=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16582);